/^iMiilciSfMi|| ^::WITH APEBICATIONS J§ J - tS ^^W?* i V,,- . •^ "- i'?". m' BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 M..^.^.s..X9.n ' rC.:i?^.z:. 9963 Cornell University Library The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031253903 Cornell University Library arV19559 The calculus for engineers and physicist 3 1924 031 253 903 olin,anx THE CALCULUS FOB ENCtTNEERS ANT) PHYSTflTRTS NET BOOK.— This book is supplied to the Trade on terms which will not allow of Discount to the Public. CHARLES GRIFFIN & CO., LTD, THE CALCULUS FOB ENGINEERS AND PHYSICISTS. GRIFFIN'S ENGINEERING PUB LICATIONS, By Pkof. R. H. smith. MEASUREMENT CONVERSIONS (English and French): 43 GRAPHIC TABLES OK, DIAGE.AMS, ON 28 PLATES. Showing at a glance the Mutual Conversion of Measurements in Different Units ot Lengths, Areas, Volumes, Weights, Stresses, Densities, Quantities of Work, Horse Powers, Temperatures, etc. For the use of Engineers, Surveyors, Architects, and Contractors. In 4to Boards. 7a. fid. In Crown 8vo. Handsome Cloth. Fully Illustrated. 6s. net. PRACTICAL CALCULATIONS FOR ENGINEERS. By CHARLES E. LARARD, A,M.Inst.C.E., M.I.Meoh.E., Wh.Exh., Head of the Mechanical Engineering Department at the Northampton Institute, London, E.G. And H. A. GOLDING, A.M.InstC.E. "We heartily recommend it both to young and old engineers, and students."— EngvMering Times. SIXTH Edition. Folio, strongly half. bound. 2l8. TRAVERSE TABLES: Computed to Four Places of Decimals for every Minute of Angle up to 100 ot Distance. For the use of Surveyors and Engineers. By RICHARD LLOYD GURDEN, Authorised Surveyor for the Governments of New South Wales and Victoria. In Large Crown Svo. Handsome Cloth. 4s. ed. net. THE THERMO-DYNAMIC PRINCIPLES OF ENGINE DESIGN. By LIONEL M. HOBBS, Engineer-Lieutenant, E..N. ; Instructor in Applied Mechanics and Marine Engine Design at the Royal Naval College, Greenwich. " The whole subject has been very methodically dealt with throughout A standard reference." — Practical SngiTieer. Third Edition. Pocket Size, Leather Limp, with Gilt Edges and Rounded Comers, on Special Thin Paper, with Illustrations, pp. i-xii+834. Price 18s. net. (THE NEW "NYSTROM") THE MECHANICAL ENGINEER'S REFERENCE BOOK. A Handbook of Tables, Formulso and Methods for Engineers, Students, and Draughtsmen. By HENRY HARRISON SUPLEE, B.Sc, M.E., " We feel sure It will be of great service to mechanical engineers." — Engineering, WORKS BY W. J. MACQUORN RANKINE, LL.D., F.R.S. A MANUAL OF APPLIED MECHANICS. SEVENTEENTH EDITION. 12s. 6d. A MANUAL OF CIVIL ENGINEERING. TWENTY-THIBD EDITION. 16s. A MANUAL OF MACHINERY AND MILLWORK. SEVENTH EDITION. 12s. 6d. A MANUAL OF THE STEAM-ENGINE AND OTHER PRIME MOVERS. SEVENTEENTH EDITION. 12s. 6d. USEFUL RULES AND TABLES. SEVENTH EDITION. 10s. 6d. A MECHANICAL TEXT-BOOK. FIFTH EDITION. 9s. LONDON: CHARLES GRIFFIN & CO., LIMITED, EXETER STREET, STRAND. THE CALCULUS ENGINEERS AND PHYSICISTS: INTEGRATION AND DIFFEEENTIATION, WITH APPLICATIONS TO TECHNICAL PEOBLEMS ; CLASSIFIED KEFERENCE TABLES OF INTEGRALS AND METHODS OF INTEGRATION. BY ROBERT H. SMITH, ^SSOO. M.LO.E., M.I.MEOH.E., M.I.EL.£.,~WH. SOH., MEM. OBDEK MEIJl, EM. PKOFESSOE OF ENOINEERINS. SECOND EDITION. REVISED AND ENLARGED. WITB DIAGRAMS. LONDON: CHARLES GRIFFIN AND COMPANY, LTD. EXETER STREET, STRAND. 1908. [All rights reserved.] -T PREFACE TO SECOND EDITION. The following remarks appeared in the Preface to the First Edition of this book : — "This work aims at the presentation of two leading features in the study and application of the higher mathematics. In the first place, the development of the rationale of the subject is based on essentially concrete conceptions, and no appeal is made to what may be termed rational imagination extending beyond the limits of man's actual physical and physiological experience. Thus no use is anywhere made of series of infinite numbers of things or of infinitely small quantities. The author believes that the logical development is both sound and complete without reference to these ideas. "In the second place, a set of Eleven Classified Tables of Integrals and Methods of Integration has been arranged in such manner as seemed best adapted to facilitate rapid reference, and thus relieve the mind engaged in practical mathematical work of the burden of memorising a great mass of formulas. " Critics who are schoolmen of the pure orthodox mathematical faith may find it hard to reconcile the ideas that have with them become innate, with some of the methods, and possibly some of the phraseology, here adopted. "We only ask them to remember that there is arising a rapidly increasing army of men eagerly engaged in the development of physical research and in the industrial applications of scientific results, with the occasional help of mathematical weapons, whose mental faculties have been wholly trained by continuous contact with the hard facts of sentient experience, and who find great difficulty in giving faith VI PREFACE TO SECOND EDITION. to any doctrine which lays its hasis outside the limits of their experiential knowledge." Experience in the use of the book since its first publication has confirmed the author in his belief that the basis upon which its treatment of the Calculus is built is 'sound, rational, logical, and that its form affords an easy and rapid method of acquiring power to apply, correctly and safely, the higher mathematics to technical problems. The method is good for technicians and physicists because it is easy and rapid. Ease and rapidity would Tae funda- mentally damning faults if it were illogical, or if it did not grow from the roots of the realities of the subject-matter. If it were illogical, it would be destructive of the intellectual training of the student. No illogicality has been discovered inthe course of a narrow criticism undertaken during the revision for this second edition. If it be throughout correctly logical, the swifter the habit of logical thinking to which the student is trained, the better for his intellectual growth. But the special virtue of the method is that this intellectual growth in mathematical power is from beginning to end fed by contemplation of the mechanical and physical facts the reality and truth of which are already parts of his familiar mental consciousness ; his primary knowledge of which is, indeed, often vague and uncritical, and which he now learns to analyse into strictly definite ideas. If this habit of correlating mathematical thinking with external observation become a con- firmed one, then his mental activity, both in logical analysis and in observation of external facts, must automatically develop con- tinuously and permanently after his formal study of mathematics has ceased. It is only by virtue of this habit that mathematical knowledge becomes of use in physics and technical engineering. The author has no fault to find with the older methods of study of transcendental mathematics, provided always that they be followed only by the select few who by temperament and choice are destined to make pure mathematics their one and only field of lifelong activity. This special kind of activity may be useful to the progress of humanity, and, although the methods are old, they develop year by year in the schools in new directions and arrive at new results. But it is only a very few specially constituted minds which are adapted to pursue these studies successfully. PREFACE TO SECOND EDITION. Vll What needs to be recognised is that it is bad training for the many not so constituted, and — what is of the most urgent impor- tance — that mathematicians of this stamp are unsuited to be, and indeed incapable of being, teachers of technical mathematics. In the revision for this new edition the work has been very carefully searched for errors. Those that have been discovered, chiefly in the cross-references between Parts I. and II., have been rectified. It is hoped that the volume is now practically free of error. Considerable additions have been made, mostly in the form of Appendices. These deal for the most part with new applications, the original work of the author, to specially important technical problems, and particularly to the problems of economy in con- struction. They include, also, additions to Part II. in the Eeference Tables of Integrals. In the course of new applications to technical work, general forms of integration which are either new or whose frequent practical utility is novel, demand a place in such Reference Tables. Both in the development of Electrical Engineering and in the stricter application of scientific method to Mechanical Design, this process of development is almost con- tinuous and inevitable. ROBERT H. SMITH. 1908. TABLE OF CONTEIfTS. PAET I. CHAPTER I.— INTRODUCTORY. 1. Integration more useful than Differentiation, . . 1 2. Method of the Schools, . 1 3. Rational Method, . , 1 4. Active Interest in the Study, 2 5. Object of present Treatise, . 2 6. Clumsiness of Common Modes of Engineering Analysis, 7. Graphic Method, . 8. Illustrations, 9. Classified List of Integrals, 10. Scope of Part I., . CHAPTER II.— GENERAL IDEAS AND PRINCIPLES, ALGEBRAIC AND GRAPHIC SYMBOLISM. X dependent on x, . ,11 Nature of Derived Functions, 11 Variation of a Function, . 12 Scales for Graphic Symbol- ism, . . . .12 Ratios in Graphic Delinea- tions, . . . .12 Differential Coefficient, a;-6radient, or X', . . 14 Scale of X', . . . . 14 Sign of X' 15 Snbtangent and Subnormal, . 16 Scale of Diagram Areas, . 17 Table of Scales, ... 17 Increments, . . . .18 Increment on Infinite Gra- dient 18 Integration, . . .19 Increment Symbols, . . 19 Integration Symbols ; Limits of Integration, . . 19 Linear Graphic Diagrams of Integration, . . .21 The Increment deduced from the Average Gradient, . 22 11. Meaning of a " Function," . 5 30. 12. Ambiguous Cases, 6 31. 13. Inverse Functions, 6 32. 14. Indefiniteness of a Function 33. in Special Cases, 6 15. Discontinuity, 6 34. 16. Maxima and Minima, 7 17. Gradient or Differential Co- 35. efficient, . . . . 7 18. Gradients at Maxima and 36. Minima, . . . . 7 37. 19. Change of Gradient, 8 38. 20. Zero Gradients, . 8 39. 21. Discontinuity or Break of 40. Gradient, 8 41. 22. Infinite Gradient, 9 42. 23. Meaning of a "Function," . 9 24. Horse-power as a Function 43. of Pressure, 9 44. 25. Function Symbols, 10 45. 26. Choice of Letter-Symbols, . 10 27. Particular and General Sym- 46. bols, . . . . 10 28. X, y, and z 11 47. 29. Functions o(x, . 11 CONTENTS. CHAPTER II.— continued. 48. 50. 51. 52. S3. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 70. 71. 72. 73. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. Area Graphic Diagram of In tegration, Oiminisbing Error, Integration through "In finite " Gradient, Change of Form of Integral, Definite and Indefinite IntC' gration, . Integration Constant, PAGE 22 22 23 24 24 24 PAGE 54, 55. Meaning of Integration Constant, . . .25 56. Extension of Meaning of In- tegration, . . .27 57. Integration the Inverse of Differentiation, . . 27 58. Usual Method of finding New Integrals, . . .27 CHAPTER III.— EASY AND FAMILIAR EXAMPLES OF INTEGRATION AND DIFFERENTIATION. Circular Sector, . Constant Gradient, Area of Expanding Circle, . Rectangular Area, Triangular Area, First and Second Powers of Variable, Integral Momentum, . Integral Kinetic Energy, Motion Integrated from Velocity and Time, . Motion from Acceleration and Time, Bending Moments, Volume of Sphere, Volume of Expanding Sphere, Volume of Expanding Pyra- mid, Stress Bending Moment on Beam, .... 36 74, Angle-Gradients of Sine and Cosine and Integra- tion of Sine and Co- sine, .... 75. Integration through 90°, 76. Spherical Surface, 77. Spherical Surface Integrated otherwise, 78. Angle-Gradients of Tangent and Cotangent and Inte- gration of Squares of Sine and Cosine, Gradient of Curve of Recipro- cals, .... a;-Gradient of Xx and inverse Integration. Formula of "Reduction," 81. x-Gradient of X/x and In- verse Integration, , 79, 80, CHAPTER IV.— IMPORTANT GENERAL LAWS. Commutative Law, Distributive Law, Function of a Function, Powers of the Variable ; Powers of Sine and Co- sine, .... Reciprocal of a Function, Product of Two Func- tions, .... Product of any number of Functions, Reciprocal of Product of Two Functions, Reciprocal of Product of any number of Func- tions Ratio of two Functions, 92. Ratio of Product of any number of Functions to Product of any number of other Functions, 93. Theory of Resultant Error, 94. Exponential Function, 95. Power-ljrradient of Expo- nential Function, . 96. Natural, Decimal and other Logarithms, . 97. Number-Gradient of Loga- rithm, .... 98. Relation between different Log-" systems," . 99. Base of Natural Logs, 100. LogarithmicDiffereutiation, 101. Change of the Independent Variable, 37 38 39 40 41 43 44 52 52 S3 54 55 55 56 57 57 58 CONTENTS. XI CHAPTER v.— PAKTICULAE LAWS. PAKE 102. Any Power of the Variable, 69 103. Any Power of the Variable by Logarithms, 104. Diagram showing Integral of x-^ to be no real excep- tion, .... 105. Any Power of Linear Function, 106. Reciprocal of any Power of Linear Function, . 107. Ratio of two Linear Functions, 108. Ratio of two Linear Functions, General case, 109. Quotient of Linear by Quad- ratio Function, 59 60 61 61 61 61 62 112, 113, 114. 122. 110. Indicator Diagrams, . 111. Graphic Constructions for Indicator Diagrams, Sin-'a; and [i^ - x^)~i, (1 -as")* Integrated, or Area of Circular Zone, . . a!(r^ - Q?)'^ Integrated, 115. (^±»^)~1 Integrated, 116. x-\r^-x')-'' Integrated, . 117. Log X Integrated, 118. Moment and Centre of Area of Circular Zone, . 119. (■fi+x^)-^ Integrated, 120. (r=-a?)-' Integrated, 121. Hyperbolic Functions and their Integrals, CHAPTER VI.— TRANSFORMATIONS AND REDUCTIONS. Derivative PAGE 62 123. 124, 125, to clear of Change of Variable, Substitution Roots Quadratic Substitution, Algebraico - Trigonometric Substitution, 126. Interchange of two Func- tions, .... 127. Interchange of any number of Functions, 128. General Reduction in terms of second Differential Coefficient, . . 129. General Reduction for X"", . 71 130. General Reduction of a»'X'', 131. Conditions of Utility of 71 same 72 132. Reduction of a?"(a-Fia?')'-, . 133. Reduction of r^^ power of 72 series of any powers of x, 134. Special Case, 73 135. 136. Trigonometrical Reduc- i tions, .... 73 137. Trigonometrico - Algebraic I Substitution, I 138, Composite Trigonometrical 73 I Reduction, 64 67 67 68 69 69 70 70 74 74 74 75 76 76 77 78 78 CHAPTER VII.— SUCCESSIVE DIFFERENTIATION AND MULTIPLE INTEGRATION. 139. The Second rc-Gradient, 140. Increment of Gradient, 141. Second Increment, 142. Integration of Second lucre ment, 143. Graphic Delineation, . 144. Integration of Second Gradient, . 145. Curvature,. 146. Harmonic Function of Sines and Cosines, . 147. Deflection of a Beam, . 148. Double Integration of Sine and Cosine Function, 149. Exponential Function, 80 81 82 82 84 84 86 86 150. 151. 152. 153. Product and Quotient of two or more a;- Functions, Third and Lower x-Gradients and Increments, Rational Integral a-Func- tions, .... Lower K-Gradient of Sine and Exponential Func- tions 154, 155. General Multiple Inte- gration, .... 156. Reduction Formulae, . 157. Graphic Diagram of Double Integration, . 158. Graphic Diagram of Treble Integration, . 87 87 89 90 91 91 xu CONTENTS. CHAPTER VIII.— INDEPENDENT VARIABLES. 159. Geometrical Illustration two Independent Vari- ables, . . .' . Equation between Indepen- dent Increments, . Equation between Inde- pendent Gradients, Constraining Relation be- tween three Variables, . Equation of Contours, General x, y, F(a^) Nomen- clature, .... 165. Two Functions of two Inde- pendent Variables, . 166. Applications to p, v, t and

{x), . . 132 217. XW=/i;a), . . . .132 218. X(»)=/<;X):XW=&X,. . 133 219. X"=yi;X) 133 220. X('')=/i;X(''-i)), . . .134 221. XW=/i;XI"-2)), . . .134 222. Xx" = (?Xy", . . .134 APPENDICES. PASB PAOB A. Time-Rates, . . .135 6. Successive Reduction For- B. Energy-Flux, . . .135 mula, , . . . 142 0. Moments of Inertia and Bend- H. Economic Proportions of I- ing Moments, . . . 136 Elimination of Small Re- sections, . . . . 143 D. I. Economic Design of Turbines, 144 mainders, . . .137 K. Commercial Economy, 147 E. Indicator Diagrams, . .138 L. Indeterminate Forms, . 150 F. Recurrent Harmonic and Exponential Functions, . 140 THE CALCULUS FOR ENGINEERS. CHAPTEE I. INTKODaCTOET. 1. Integration more useful than Differentiation. — In phy- sical and engineering investigations the Integral Calculus lends more frequent aid than does the Differential Calculus, and the problems involving the Integral are more often of a practically important type than those requiring the Differential Calculus alone in their solution. But the ordinary student of mathematics never reaches even an elementary knowledge of Integration until he has mastered all but the most recondite portions of the science of Differentiation. It seems a priori irrational, and contrary to a hberal conception of educational policy, to teach the higher mathematics in a manner so contrary to almost self-evident utility. Adherence to this the orthodox method of teaching in the Schools and Universities is, no doubt, responsible for the persistent unpopularity of this branch of knowledge and intellectual training among the classes devoted to practical work. 2. Method of the Schools. — It must be admitted that no great progress can be made in Integration without help from the results obtained by Differentiation. Therefore, so long as the two are taught as distinct subjects, by the aid of separate text-books, it is a distinct convenience to the teachers to finish off one before entering upon the other. If they be thus separated into two successive periods of study, it becomes a practical necessity to give Differ- entiation the priority in point of time. 3. Rational Method. — Still, it by no means follows that the whole of the science of Differentiation must be known before any of that of Integration can be explained, thoroughly mastered, and A I THE CALCULUS FOR ENGINEERS. utilised. The ordinary system of teaching the subject forces the practical student to spend on Differentiation an amount of time altogetherneedless for his professional objects before ha enters upon Integration. Much of the former he will never use. The latter, from the very beginning, will supply him with abundant problems of immediate interest and importance in his own special work, and wUl, moreover, furnish him with a powerful engine that will enor- mously lighten the difficulties of his own professional subjects and make his progress in these tenfold more rapid. Let it be noted, also, that very frequently the reasoning used to find an integral is essentially the same as that used to find the inverse difierential. It is thus pure waste of time to go thiough this reasoning twice over. Once understood, it leads to the simul- taneous recognition of the two inverse results, both of them, it may be, eminently useful. Therefore, as far as practicable, the study of Differentiation and Integration ought to be pursued 'pari passu. 4. Active Interest in the Study. — In modern education, in which such large demands are made upon the intellectual energies of the pupil, the necessity of the stimulus of a real active interest, opening out easily recognised prospects of broadening and deepening knowledge and of utilitarian advantage, ought to be conceded in the freest and most liberal fashion. Moreover, it is right to lead the pupU along the easiest road, provided it be a legitimate one. The thoroughness of the training he receives in habits of sound, trustworthy scientific thought depends more upon the length of time he is guided within the limits of correct method, and less upon whether he travels a short distance on a rugged and difficult path or a long distance upon a plainer and smoother route. 5. Object of Present Treatise. — The object of the present treatise is to introduce the student at once to the fundamental and important uses of the Integral Calculus, and incidentally to those of much of the Differential Calculus. This we desire to do in such a way as to stimulate a growing desire to progress always further in a branch of science which soon shows itself capable of supply- ing the key to so many practical investigations. 6. Clumsiness of Conunon Modes of Engineering Analysis. — At the present time our technical text-books are loaded with tedious and clumsy demonstrations of results that can be obtained " in the twinkling of an eye " by one who has grasped even only the elements of the Calculus. These demonstrations are supposed to be " elementary." They are not really so ; each of them really contains, hidden with more or less skUl, identically the same reasoning as that employed in establishing the Calculus formulas applicable to the case in hand. They are, in fact, simply laboured INTRODUCTORY. 3 methods of cheating the student into using the Calculus without his knowing that he is so doing. There is no good reason for this. The elements of the Calculus may be made as easy as those of Algebra or of Trigonometry. More good, useful scientific result can be obtained with less labour by the study of the Calculus than by that of any other branch of mathematics. 7. Graphic Method. — Much of the Calculus can be rigorously proved by the graphic method ; that is, by diagram. This method is here used wherever it offers the simplest and plainest proof ; but where other methods seem easier and shorter they are preferred. The present book is strictly confined to its own subject ; and, wherever it is necessniy, the results proved in books on Geometry, Algebra, Trigonometry, etc., are freely made use of; employing always, however, the most elementary and most generally known of these results as may be sufficient for the purpose. 8. Illustrations. — Everywhere the meaning and the utility of the results obtained are illustrated by applications to mechanics, thermodynamics, electrodynamics, problems in engineering design, etc., etc. < 9. Classified List of Integrals.— XThe part of the book which is looked upon by its authors as the roost important and the most novel is the last, namely, the Classified Reference List of Integrals. This is really a development of a Classified List of Integrals which one of the authors made twenty years ago to assist him in his theoretical investigations, and which he has found to be con- tinuously of very great service. He has never believed in the policy of a practical man's burdening his memory with a load of theoretical formulas. Let him make sure of the correctness of these results, and of the methods by which they were reached. Let him very thoroughly understand their general meaning, and especially the limits of their range of applicabiHty ; let him recognise clearly the sort of problem towards the solution of which they are suited to help ; let him practise their application to this sort of problem to an extent sufficient to make him feel sure of himself in using them in the future in the proper way. Then let him keep notes of these results in such a manner as will enable him to find them when wanted without loss of time; and let him particularly avoid wasting his brain-power by preserving them in his memory. The more brain-power is spent yi^emorising, the less is there left for active service in vigorous and wary application in new fields to attain new results. Formulas have a lamentable characteristic in the facihty they offer for wrong application. A formula fixed perfectly in the memory, and the exact meaning and correct mode and limits of whose application are imperfectly understood, is a 4 THE CALCULUS FOR ENGINEEES. pure source of misfortune to him who remembers it. It is infinitely more important to cultivate the faculty of cautious and yet ready use of formulas than to have the whole range of mathematical formulas at one's finger ends ; and this is also of immensely greater importance to the practical man than to keep in mind the proofs of the formulas. To obviate the necessity of such memorisation the "Classified Eef erence List of Integrals " has been constructed in the manner thought most likely to facilitate the rapid finding of whatever may be sought for. The results are not tabulated in " rational " order, that is, in the order in which one may be logically deduced from preceding ones. They are classified, firstly, according to subject, e.g., Algebraical, Trigonometrical, etc., etc., and under each subject they are arranged in the order of simplicity and of most frequent utility. A somewhat detailed classification has been found desir- able in order to facilitate cross-references, the free use of which greatly diminishes the bulk of the whole list. The shorter such a list is made, the easier is it to make use of., 10. Scope of Prefatory Treatise. — This treatise does not prove all the results tabulated in the " Keference List." The latter has been made as complete as was consistent with moderate bulk, and includes all that is needed for what may be described as ordinary work, that is, excluding such higher difficult work as is never attempted by engineers or by undergraduate students of physics. The treatise aims at giving a very thorough imderstanding of the principles and methods employed in finding the results stated concisely in the " Eeference List " ; proofs of all the fundament- ally important results ; and, above all, familiarity with the practical uses of these results, so as to give the student confidence in his own independent powers of putting them to practical use. The last chapter on the Integration of Differential Equations ought to aid greatly in pointing out the methods of dealing with various classes of problems. The niath chapter, on Maxima and Minima, is per- haps more illustrative than any other of the great variety of very important practical problems that can be solved correctly only by the aid of the Calculus. GENERAL IDEAS AND PRINCIPLES. CHAPTEE II. GENBBAL IDEAS AND PEINCIPLBS— ALGEBRAIC AND GRAPHIC SYMBOLISM. 11. Meaning of a " runction."— Suppose that a section be made through a hilly bit of country for some engineering puipose, such as the making of a highway, or a railway, or a canal. The levels of the different points along the section are obtained by the use of the Engineer's Level, and the horizontal distances by one or other of the ordinary surveying methods. Let fig. 1 be the plot- / !*■ Z Fig. 1. ting on paper of the section. According to ordinary practice, the heights would be plotted to a much opener scale than the horizontal distances ; but in order to avoid complication in a first illustration, we will assume that in fig. 1 heights and distances are plotted to the same scale. Each point P on this section is defined strictly by its level h and its horizontal position I. The former is measured from some conveniently chosen datum level. The latter is measured from any convenient starting-point. These two are called by mathematicians the co-ordinates of the point P on the curve ABC, etc. Eor each ordinate I there is one defined value of the co-ordi- nate h ; except throughout the stretch MN, where a break in the curve occurs. Putting aside this exception, the height h is, when this strictly definite relation exists, called in mathematical language a " function "oil; or Height = Function of Horizontal Distance, or, more simply written in mathematical shorthand, A = F(Z). 6 THE CALCULUS FOE ENGINEERS. 12. Ambiguous Cases. — As seen from the dotted line drawn horizontally through P, there are three points on the section at the same level. Thus the statement that Distance = Function of Height or l=f{h) must be understood in a somewhat different sense from the first equation : namely, in the sense that, although for each height there correspond particular and exactly defined distances, still two or more such distances correspond to one and the same height, so that, if nothing but the height of a point were given, it would remain doubtful which of two or three horizontal positions it occupied. This ambiguity can only be cleared away by supplying special in- formation concerning the point beyond that contained in the equation. 13. Inverse Functions. — The two formulas h='E{l) and i=m are simply two different forms into which the relation between h and I, or the equation to the curve, can be thrown. The first form may be called the solution of the equation for h; the second the solution of the equation for I. The functions F( ) and /( ) are said each to be the " inverse " of the other. An inverse function is frequently indicated by the symbol - 1 put in the place of an index. For example, if s be sin a, then the angle a may be written sin 'h. Or if I be the logarithm of a number N, or Zi=log N; then N = log"'/, which ex- pression means that " N is the number whose log is I." 14. Indefiniteness of a Function in Special Cases. — The stretch of ground from E to S is level. Here the value of h corre- sponds to a continuously varying range of values of l. For this particular value of h, therefore, we have between certain limits in- definiteness in the solution for I. If there were under the point J a stretch of perfectly vertical cliff, then for the one value of / to this cliff the solution for h would be similarly indefinite between the limits of level at the foot and at the top of the cliff. 15. Discontinuity.— From M to N there is a break in the curve. In such a case mathematicians say that h is a discontinuous function of I ; the discontinuity ranging from M to N. GENERAL IDEAS AND PKINOIPLES. / 16. Maxima and Minima. — From A to C the ground rises ; from C to E it falls. At C we have a summit, or a maximum value of h. This maximum necessarily comes at the end of a rising and the beginning of a falling part of the section. Evidently the converse is also true, viz., that after a rising and before the following falling part there is necessarily a maximum, provided there he no discontinuity between these two parts. There is another maximum or summit at K. The ground falls continuously from C to E, and then rises again from E onwards. There is no discontinuity here, and E gives, therefore, a lowest or mimmum value of h. This ■ necessarily comes after a falling and before a rising part of the section ; and between such parts there necessarily occurs a minimum, if there be no discontinuity. We have here assumed the forward direction along the section to be from A towards the right hand. But it is indifferent whether we call this or the reverse the forward direction as regards the distinction between maximum and minimum points. 17. Gradient or Differential CoeflBcient. — Each small length of the section has a definite slope or gradient. Engineers always take as the measure of the gradient the ratio of the rise of the ground between two points near each other to the horizontal dis- tance between the same points. This must be carefully distin- guished from the ratio of the rise to distance measured along the sloping surface. This latter is the sine of the angle of inclinar tion of the surface to the horizontal ; whereas the gradient is the tangent of the same angle of inclination. This gradient is the rate at which h increases with I. It is, in the present case, what is called a space rate, or length rate, or linear rate, because the increase of h is compared with the increase of a length I (not because h is a length, but because Z is a length). If at the point Q the dotted line Qg be drawn touching the sec- tion curve at Q, then the gradient at Q is the tangent of the angle Q2O. The touching line at point P on the downward slope cuts 00' at p, and the tangent of PpO' is negative. It equals the tangent of PpO with sign reversed. In the language of the Calculus this gradient is called the Dif- ferential CoefBcient of h with respect to I. Taking the forward direction as from A towards the right hand, the gradient is upward or positive from A to the summit C ; downward or negative from C to the minimum point E; positive again from E to K, and negative from K to M. From N to E it is positive, and along ES it is zero. 18. Gradients at Maxima and Minima. — At each maximum 8 THE CALCULUS FOR ENGINEERS. and minimum point (C, E, K) the gradient is zero. At each maxi mum point (C, K) it passes through zero from positive to negative. At a minimum point (E) it passes through zero from negative to positive. At H there is also level ground, or zero gradient. Here, how- ever, there is neither maximum nor minimum value of h. This point comes between two rising parts of the section: there is a positive gradient both before and after it. Although, therefore, we find zero gradient at every maximum and at every minimum point, it is not true that we necessarily find either a maximum or a minimum wherever there is zero gradient. , 19. Change of Gradient. — On the rising part of the ground it becomes gradually steeper from A to B ; that is, the upward gra- dient increases. Otherwise expressed, there is a positive increase of gradient. From B to C, however, the steepness decreases; there is a decrease of gradient, or the variation of gradient is negative (the gradient itself being stiU positive). Thus the variation of gradient being positive from A to B and negative f rdm B to C, passes through' the value zero at B, the point ■j^ere the gradient itself is a maximum. From C to D the gradient is negative, and becomes gradually steeper ; that is, its negative value increases, or, otherwise expressed, its variation is negative. From D to E the gradient is negative, but its negative value is decreasing, that is, its variation is positive. Thus at D the variation, or rate of change of gradient, changes from negative to positive by passing through zero, and at this point D we have the steepest negative gradient on this whole slope CE. The steepest negative gradient, of course, means its loivest value. Thus at D we have a minimum value of the gradient. 20. Zero Gradients. — The distinction between the three parts C, E, and H, at all of which the gradient is zero, becomes now clear. At C the variation of the gradient is negative, and this gives a maximiun height. At E this variation is positive, and here there is a minimum height. At H this rate of variation of the gradient is zero, and here, although the gradient be zero, there is neither maximum nor minimum height. 21. Discontinuity or Break of Gradient.— Wherever there is a sharp corner in the outhne of the section, as at I, J, E, S, T, U, there is a sudden change or break of gradient. This means that at each of these points there is discontinuity of gradient ; and the above laws will not apply to such points. Wherever there are points of discontinuity, either in the curve itself or in its gradient, special methods must be adopted in any investigations that may be undertaken in regard to the character- GENERAL IDEAS AND PRINCIPLES. 9 istics of the law connecting the ordinates. The methods applicable to the continuous parts of the curve may, and usually do, give erroneous results if applied to discontinuous points. 22. Infinite Gradient. — Under J the face being vertical, the gradient is commonly said to be " infinite." At each of the sharp points I, J, K, S, T, U, the variation of gradient being sudden, the rate of variation of gradient becomes "infinite." More cor- rectly expressed, there exists no gradient at J ; and at I, J, R, etc., there are no rates of variation of gradient. 23. Meaning of a " Function." — The symbolic statement /i = F(Z) is not intended to assert that the relation between I and h is expressible by any already investigated mathematical formula, whether simple or complicated. For example, in fig. 1 the said relation would be extremely difficult to express by any algebraic or trigonometric formula. Equally complicated would be the law expressing the continuous variation of, for example, the horse- power of a steam-engine on, say, a week's intermittent running ; or that connecting the electric out-put of a dynamo when connected on to a circuit of variable and, perhaps, intermittent conductivity. Yet separate short ranges of these laws may in many cases be approximated to by known mathematical methods ; and even when this is not possible, many very interesting, important, and prac- tically useful special features of the general law may be investi- gated by mathematical means, without any exact knowledge of the full and complete law. Thus without making any reference to, or any use of, the exact form of the function F( ) in the equation appKcable to fig. 1, we have already been able to point out many important features of the law it represents. 24. Horse-power as a Function of Pressure. — Again, although the actual running of, say, a steam-engine from minute to minute varies with many changes of condition, still, if we choose to investi- gate the separate effect of one only of these changes, for instance, change of initial pressure, it may be found fairly simple. Thus we may write Horse-power = Function of Initial Pressure, or where p is the pressure. This means that any change of pressure changes the horse-power ; and to investigate the separate effect of change of pressure on horse-power, we consider all the other con- 10 THE CALCULUS FOR ENGINEERS. ditions to remain (if possible) constant, while the pressure changes. Some other conditions may themselves necessarily depend on the pressure, and these, of course, cannot be assumed to remain con- stant. For example, the cut-oif may be supposed to remain con- stant. But the amount of initial steam condensation in the cylin- der depends partly on the initial pressure, and it cannot, therefore, be assumed a constant in the equation HP = ^( p). Similarly, the HP may be considered as a function of the speed, it alone being varied while aU other things are kept constant. Or the HP may be taken as a function of the cut-off, the initial pressure, the speed, and everything else being kept constant, wMle the cut-off is varied. 25. runction Symbols. — When different laws connecting certain varying quantities have to be considered at the same time, different symbols, such as ^(O. /(O. <^(^). •/'(O. are used to indicate the different functions of I referred to. 26. Choice of Letter-Symbols. — In fig. 1 we have used I to represent a distance, because it is the first letter in the word " Zera^^A,," and similarly A to represent " heiffht." It is very desir- able when letter-symbols have to be used, to use such as readily call to mind the nature of the thing symbolised. Especially in practical applications of mathematics, and more particularly when there is any degree of oompUcation in the expressions involved, is the adherence to this rule to be strongly recommended. By keep- ing the mind alive to the nature of the things being dealt with, error is safeguarded against, and the true physical meaning of the mathematical operations and of their results are more easily grasped. Without a complete understanding of the physical meaning of the result, not only is the result useless to the practical man, but its cor- rectness cannot be judged of. If, on the other hand, the physical meaning be fully grasped, any possible error that may have crept in in the mathematical process of finding the result, is likely to be detected and its source discovered without great difficulty. 27. Particular and General Symbols. — But many mathemati- cal rules and processes have such wide application to so many entirely different physical conditions, that, in order the more clearly to demonstrate the generality of their application, mathematicians prefer to use letter-symbols chosen purposely so as to suggest only with difficulty anything endowed with special characteristics ; such as X, y, z, symbols which do not suggest to the mind any idea whatever except that of absolute blankness. It is doubtful whether this is a desirable habit in mathematical training. It seems probable that a course of reasoning might be GENEKAL IDEAS AND PRINCIPLES. 11 more firmly established in the mind of the student if he were first led through it in its concrete and particular aspect — the mind being kept riveted on one special set of concrete meanings to be attached to his symbols — and then afterwards, if need be, he may go through it again once, twice, or, if necessary, a dozen times, in order to dis- cover (if or when this be true) that the general form of the result will remain the same whatever particular concrete meanings be attached to his symbols. 28. X, y, and z. — There is one feature in the use ordinarily made of x, y, z in mathematical books which the writer thinks is a real evil. In his earlier chapters the orthodox mathematician establishes a habit of using y to indicate a function of x : he con- stantly writes y =f{x) : that is, he takes y to represent a thing dependent on x, and which necessarily changes in quantitative value when x changes. But in his later chapters he uses y and X as two independent variables, that is, as two quantities having no sort of mutual dependence on each other, the variation of either one of which has no effect whatever upon the other. This is apt to, and does, produce confusion of mind ; especially as regards the true meaning of different sets of formulas very similar in appear- ance, one referring to y and x as mutually dependent quantities, the other referring to y and x as independent variables. 29. Functions of x. — When x is used to indicate a variable quantity, any other quantity whose value varies in a definite way with the varying values of x, may be symbolically represented in any of the following ways : — ¥{x), fix), 4>{x), fix), xix), and X, ^ or H. The last forms, X, etc., are for shortness and compactness as con- venient as y, and are more expressive. They will be used chiefly in connection with x in the following pages. 30. X dependent on x. — X may mean a function which is capable of being also changed by changing the values of one or more other quantities besides x ; but in so far as it is considered as a function of x, consideration of these other possible changes is eliminated by supposing them not to occur. This is legitimate because these other elements which go to the building up of X do not necessarily change with x. All elements involved in X, which necessarily change with x, are to be expressed in terms of x, and their variation is thus taken account of in calculating the variation of X. 31. Nature of Derived Functions. — In dealing with functions of this kind, mathematicians call x the "independent variable," a somewhat unhappy nomenclature. X and x are in physical reality mutually dependent one on the other. In the mathematical 12 THE CALCULUS FOR ENGINEERS. formula, however, X being expressed in terms of x, it is considered as being derived from, or dependent upon, x ; the various values of X being calculated from those of x, and the changes in X being calculated. from the changes in x. Thus it should be Ijorne in mind that the dependence of the one on the other suggested in the commonly used phrase " independent variable " is purely a matter of method of calculation, and not one of physical reality. 32. Variation of a Function. — Similarly Y may be used to indicate a derived or " dependent " function virhose value depends only upon constants and upon the variable y. Or L may be made to denote a derived function depending only on constants and on the variable /. 33. Scales for Graphic Symbolism. — Those readers of this treatise who are engineers must, from practice of the art of Graphic Calculation, be famihar with the device of representing quantities of all kinds by the lengths of lines drawn upon paper, these lengths being plotted and measured to a suitable Scale. So long as the quantity of a function is its only charapteristic with which we are concerned, each quantity can always be repre- sented by the length of a line drawn in any position and in any direction on a sheet of paper, the scale being such that 1 inch or 1 millimetre of length represents a convenient number of units of the kind to be represented. In "Graphic Calculation " we very commonly represent on the paper also the two other characteristics of position and direction of the things dealt with; but in the Differential and Integral Calculus, so far as it is dealt with in this treatise, we are concerned only with quaniity. It is convenient to draw all lines representing the various values of the same kind of thing in one direction on the paper. Thus we may plot off all the a;'s horizontally and the corresponding X's vertically. If, when the magnitude of x is varied continuously {i.e., vrithout break or gap), the change of X be also gradual and continuous, there is obtained by this process a continuous curve which is a complete graphic representation of the law connecting X and X. The student ought at the outset to understand fuUy the nature of this kind of representation. It is clear that it is in its essence as wholly conventional and symbolic as is the letter- symbolism of ordinary algebra. Spoken words, written words, and written numbers are in the same way conventional ; they also constitute systems of arbitrary symbolism. Graphic diagram repre- sentation is neither more nor less symbolic and arbitrary than ordinary language. 34. Ratios in Graphic Delineation. — The curve in flg. 2 is such a graphic delineation of a law of mutual dependence between X GENERAL IDEAS AND PRINCIPLES. 13 and X. If X te of a diiferent kind from x, it is impossible to form any numerical ratio between th.e two scales to which X and x are plotted. For instance, if the X's are ft.-lbs. and the k's feet, then the vertical scale may be, perhaps, 1" = 10,000 ft.-lbs., and the horizontal scale 1" = 10 ft. ; but the number — i— — , or 1000, is not a pure ratio between the two scales. But in physics we have relations between things of different kinds, which are called physical ratios. It is only by use of such physical ratios that derived T:, dx ^ "Z. _dx *__!/ Fig. 2. quantities are obtained. Thus the physical ratio between a number of ft.-lbs. and a number of feet, or ft.-lbs./ft., is a number of lbs. The ratio is of an altogether different kind, in this case lbs., from either that of the dividend or that of the divisor. Now the ratio between a height and a horizontal distance on the paper is a gradient measured from the horizontal. In this example, then, a gradient would mean a number of lbs., 14 THE CALCULUS FOE ENGINEEES. and each gradient would represent lbs. to a certain scale. Con- tinuing the above example, a tangent or gradient measuring unity on the paper, i.e., the tangent of 45°, would mean ^Q'Q?;;-^'"^ 1000 lbs. 10 It. This is the scale to which gradients from axis of x are to be measured ; or Unit gradient = Unit height on paper _ ^ ^^^^ j^^, ° Unit horizontal length on paper Gradients measured from the axis of X have a reciprocal interpre- tation and are to be measured to a reciprocal scale. Thus a;/X means ft./ft.-lbs., or 1/lbs., that is, the reciprocal of a number of lbs. 35. Differential CoeflBcient, a;-Gradient or X'. — The gradient from axis of x of the curve at any point x, X, is called the " DiflFer- ential Coefficient of X with respect to x," and is symbolised by either ^^ or X'. ax The phrase " Differential Coefficient of X with respect to x " is a cumbrous one. A shorter phrase is the avgradient of X ; and as this phrase is very easily understood and definitely descrip- tive, it is used in this treatise. The gradient of the curve from the axis of X is the reciprocal of the above. It is called the DiflFerential Coefficient of x with respect to X, or the X-gradient of x ; and is shortly written -— : . aX 36. Scale of X'. — The scales of X and x are in general different ; and that of X' must always be different from either of these. The numerical relation between these scales and that of -^ may be uX thus expressed : Let the scale of the a^s be l" = s units of the x kind or quality ; )i )) -^ s ,, 1 = o ,, ,, A ,, ,, GENERAL IDEAS AND PRINCIPLES. 15 then the scale of the X"s is S X Unit gradient = tan 45° = — units of the — kind : s X and the scale of the -— = is aX Unit gradient = tan 45° = — units of the — kind. o X 37. Sign of X'.— The sign of X' is + when the slope of the curve is such as to make both X and x increase positively at the q» ^, c. Fig. 3. same time ; it is - when it makes one increase while the other decreases. Evidently -^ must always have the same sign as X'. dX. The possible variation's of X' and -^ are very fully illustrated in fig. 3. 16 THE CALCULUS FOR ENGINEEES. In fig. 3, + a; is measured towards the right and + X upwards ; negative k's are measured towards the left and negative X's down- wards. The student should follow out the variations from + through to - of hothX'and — throughout the lengths of all the four dX curves A, B, C, and D in each of the four quadrants. 38. Subtangent and Subnormal.. — In fig. 2 there are drawn three hnes from a point xX of the curve, viz., a vertical, a tangent, and a normal. These intercept on the axis of x the lengths marked T^, and N^, on the diagram. T^ is called the subtangent and Nj the subnormal. Since (by definition) the tangent has the same gradient as the curve at its touching point, evidently and X' = |.orT.4 Here T,, measured to the a!-scale, and interpreted as being of the same kind as the x's, is a true graphic representation of X/X'. But —2 is of the same kind as x/X, and, therefore, would be not of the same kind as X/N^., if N^ were measured to the a>-scale, and interpreted as of the same kind as x. Thus in order that N^, may be used as a true graphic representation of X'X, which is of the same kind as X^/x, care must be taken not to measure it to the x^cale, and not to interpret it as the same kind of thing as X. If the diagram were replotted, leaving the a!-scale unaltered, and making the X-scale more open, the paper-height of X would be increased, and the paper-gradient X' would be increased in the same proportion. It can easily be shown that the paper-length of Ta, would remain unaltered, while that of N^, would be increased in a ratio which is the square of that in which X is increased. Simi- larly if, while the X-scale is maintained unchanged, the x-scale were altered so as to increase the paper-length of x, then the paper-gradient of the curve would be decreased in the same pro- portion as a; is increased ; Tj, would be increased in the same pro- portion as a; ; N^ would be decreased in the same proportion. Thus Nj; in order to be a true graphic representation of X'X, a &ENERAL IDEAS AND PRINCIPLES. 17 quantity whose dimensions are those of XV»;, must be measured to the scale l" = §i units of the (f) kind. In fig. 2, Ta, and N^, are taken upon the ic-axis, and may be termed the P5-subtangent and a;-subnormal. If ,the curve-touching line and the normal be prolonged to cut the X-axis, they and the horizontal through the touching-point will give intercepts on the X-axis, which may be termed the X-subtangent and X-subnormal, and may be written T^ and 'R^. They are shown on fig. 2, and their proper scales are given below. 39. Scale of Diagram Areas. — An area enclosed by any set of lines upon such a diagram may be taken as the graphic repre- sentation of a quantity of the same kind as the product X.x, and must be measured to the scale, 1 sq. inch = (Ss) units of the (Xa;) kind. 40. Table of Scales. — The following is a table of interpreta- tions of the diagram. This diagram wiU be constantly used here- after for both illustrations and proofs, which latter cannot be accepted as legitimate unless the whole nature of this manner of symbolic expression be intimately understood. Table of Intbrpeetations and Scales of DiAaHAMUATio ok Graphic Eepresbntations of Dbrivativh and Derived Functions. Name. Interpretation. Symbol. Diagram Scale. Variable, .... x X 1" = s units of K kind. Function of x, . . X X 1" = S „ X ,, a;-6radient of X, Ratio of small in- crease of X to accompanying in- crease of a. X' tan 45° =^ „ XJx „ X-Gradient of x, Ratio of small in- dx tan 46° =| , „ xjX „ crease of X to dX accompanying in- crease of X. a-Subtangent, . . X/X' T. i"=| .. xy. „ 1" = S „ X „ K-Subnormal, XX' N-» X-Subtangent, ^Ift^""' Tx X-Subnormal, . . ^1^'-^ Nx I'S- -Wx,. Area xX. A 1 sq. in. =sS ,, xX ,, 18 THE CALCULUS FOR ENGINEERS. 41. Increments. — In going forward from a point on the curve a little way, a rise occurs if the gradient be iipwards. The short distance measured along the sloping curve may be resolved into two parts, one parallel to axis of x, the other parallel to axis of X. These two parts are the projections of the sloping length upon the two axes. They constitute the differences of the pairs of x and X co-ordinates at the boguining and the end of the short sloping length. These differences are designated by the Greek 8; thus, see fig. 4, 8x projection on aj-axis, ana oX ,, „ X ,, Since the gradient X' is the ratio of rise to horizontal distance throughout a short length, it is evident that SX = X'8x. If I and L be the co-ordinates, and if the gradient be called L', then this would be written 8L = L'SZ. If p and Y were the co-ordinates, the gradient being called Y', then SY = Y'Sy. 42. Increment in Infinite Gradient. — These are the direct self- evident results of the definition of gradient, or differential co- efficient. They do not, of course, apply to points where there is no gradient, that is, to sharp corners in a diagram, where the direc- tion of the diagram line changes abruptly. If the diagram line run exactly vertical at any part, then for that part X' becomes infinite, and the equation appears in the form SX = 00 X an indeterminate form. GENERAL IDEAS AND PRINCIPLES. 19 This last case corresponds to the piece of vertical cliff under point J in the section fig. 1. 43. Integration. — The general case corresponds to the gradual stepping along the other parts of this section. The length of each step is projected horizontally {81 or Sx) and vertically (8^ or 8X). The latter is the rise in level, and it equals the gradient multiphed hy the horizontal projection of the length of step. In stepping continuously from one particular point on the section to another, for instance, from A to C on fig. 1, the total horizontal distance between the two is the sum of the horizontal projections (the Si's or &'s) of all the separate steps ; and the total difference of level is the sum of the vertical projections (the Sh's, or 8L's, or 8X's) of all the separate steps. In climbing the hill, the cKmber rises the whole difference of level from A to C, step by step : the total ascent is the sum of all the small ascents made in all the long series of steps. If the distance be considerable, the number of steps cannot be counted, except by some counting instrument, such as a pedometer ; but the total ascent remains the same, whether it be accomplished in an enormous number of extremely short steps or in an only moderately large number of long strides. The mathematical process of calculating these sums is called Integration. This mathematical process is indicated by the symbol the Greek capital S, when the individual steps are of definitely measurable small size. But when the method of summation employed is such that it assumes the steps to be minutely and immeasurably small, the number of them being proportionately immeasurably large, and when, therefore, of necessity the method takes no account of, and is wholly independent of, the particular minute size given to the steps, then the symbol employed is I , which may be looked upon as a specialised form of the English capital S, the first letter of the word " sum." The result of the summation is called the Integral. 44. Increment Symbols. — The separate small portions, whose sum equals the Integral, are called the Increments or the Differ- entials. When the increments are of definitely measurable small size, they are indicated by the symbols 8a;, 8X, Sh, 8L, 8Y, etc., etc. When they are immeasurably minute, and their number corre- spondingly immeasurably large, they are indicated by the symbols dx, dX, dh, dL, dY, etc., etc. 45. Integration Symbols. Limits of Integration. — The inte gration is carried out between particular limits, such as B and C in 20 THE CALCXJLUS FOE ENGINEERS. fig. 1. These limits are sometimes written in connection with the symbols of integration, thus : c C /"C ro •2 8h , "2 SI OT dh , dl. B B J B J B If IJic be the co-ordinates of the point C, fig. 1, and Zb?% be those of the point B, then these integrals mean the same thing as {ho - h^) or {Ic - Ib) ■ The limits are above indicated in the symbol by the names only of the points referred to. The points themselves are, however, frequently indicated only by the values of their co-ordinates, and then H is customary to indicate the limits of integration oy writing at top and bottom of the sign of integration the limit-values of the variable whose increment appears in the intregral. Thus, since shAi at we have the integral of Sh between B and C expressible in the two following forms : If particular points be indicated by numbers, the symbolism be- comes somewhat neater. Thus the integral of SX between the points 1 and 2 of the x, X curve at which points the ordinates may be called x^ x^, and the co-ordinates Xj X2, is ■X, r^2 = X'a 'x, Or, again, if it were convenient to call the two Umiting values of X by the letter-names a and b, then the same would appear as 'b X'dx. a Or, if the limiting values of x were, say, 15 and 85 feet, it would be written "85 X85-Xi5= I X'dx. GENERAL IDEAS AND PEINCIPLES. 21 It must be noted that the limits which are written in always refer to values of the variable whose increment or differential appears in the integration. Thus the a and b or the 85 and 15 above mean invariably values of x, not values of X nor of X'. 46. Linear Graphic Diagrams of Integration. In figs. 5 and 6 are given two methods of graphically representing this process of integration. The first corre- sponds with the illustrations we have already employed. Here the curve xX is supposed to be built up step by step by drawing in each small stretch of horizontal length Sx at a gradient equal to the known mean gradient X' for that length. The gradient X' is supposed known for each value of X, and its mean value throughout each very small length Sx is therefore known. "With regard to this statement it should be noted that a curve does not really possess a gradient at a point, but only through- out a short length. When we speak of the slope of a curve at a point, what we really mean is the slope of a minute portion Fig. 5. Fig. 6. of its length lying partly in front and partly beyond the point : that is, there is actually no difference of meaning between the phrases "the slope of the curve at the point" and "the mean gradient throughout a short length at this point." Since each increment of X, or 8X, equals X' times the corresponding in- crement of X or Sx, we have in fig. 5 all these increments of X Missing Page Missing Page 24 THE CALCULUS FOR ENGINEERS. There is, however, nothing meaningless or impossible in / dX at the same place. In fig. 1 up the vertical face under J, the 8X's, or Sh's, have the same concrete, finite meaning as they have else- where. Thus it is clearly improper to write I dX = I X'dx for this part of the integration ; the formula, which is true in general, fails under these special conditions. 61. Change of Form of Integral. — If L be a function of the variable I, and if its Z-gradient be called L', then SL = L'8Z ; and if A, be any other function of I, then | Xdl = I — , dL. When A and L' are both capable of simple expression in terms of L, the latter form of the integral may be more easUy dealt with than the former. Such a transformation of an integral is called a change of the independent variable or " substitution." * 52. Definite and Indefinite Integrals. — Sometimes the limits of the integration are not expressed in the written symbol, which then stands simply / X'dx. When thus written, it is under- stood that in the integration the variable x increases continuously up to an undefined limiting value, which is to be written x in the expanded form of the integral. In fact by / X'dx is meant | X'dx, the upper limit being any final value of the gradually increasing x. The lower limit may be written without defining the upper limit. Thus I aX'dx means (X - X^ ). If various upper limiting values of X be successively taken, the part of the integral function involving a remains unchanged. Such an integral may be written (x'dx = X + G, that is, as the sum of two terms, one of which, C, remains unchanged when the upper limit is varied, while the other, X, remains the same although the lower limit be changed. This is called the indefinite integral, and C is called the constant of integration. When both upper and lower limits are particularised, as in aX'dx, the quantity is called a definite integral. 53. Integration Constant. — To show the exact meaning of the * See Classified List, II. G. / GENERAL IDEAS AND PEINCIPLES. 25 integration constant C, compare tlie above two forms of writing the indefinite integral. The values of X being the same in both cases, it is clear that C equals ( - X^). The integration constant, therefore, depends on the imphed lower limit of w ( = a). If C be given, the implied lower limit a is thereby fixed ; and conversely, if a be given, its value determines that also of C. The indefiniteness of the indefinite integral may, therefore, be considered as due to free choice being left as to either or both limits. The part X depends on the choice of the upper Hmit, and remains indefinite so long as that is not fixed. The part C depends on the lower limit, and is indefinite until this limit is fixed. 54. Meanii^ of Integration Constant. — Figs. 7 and 8 may help to elucidate further this question of limits and of integration con- stant. In fig. 7 the same curve is drawn thrice in different posi- tions in the diagram. P'Q'E' is PQE simply raised at every point X /-! •se 1 i. Fie. 7. through the height m. P"Q"E" is the same as PQE shifted horizon- tally the distance n. Since X' is the gradient of the curve, the same values of the integral I X'c^k will be obtained from all three curves if it be taken between limits on each which give the same series of values of X'. Thus the integrals will be the same when obtained from PQE and from P'Q'E' if the same limits of x be used in each case. They will be the same from PQE and from P"Q"E" if the same limits of X be used, which wiU mean limiting values of x in P"Q"E" greater by n than those in PQE. These upward and right-hand horizontal shif tings of the curve 26 THE CALCULUS FOR ENGINEERS. are equivalent to equal downward and left-hand horizontal shif t- ings of the axes from which the co-ordinates X and x are measured. Thus the two shiftings are combined in fig. 8. Here, in order to J < n — * X / i T' / r-»-a; K - *, - H 1 1 m 1 + If ^-* ^k + mx and I (k + mx)dx = Aa; + ^mv? + C . \^ 64. First and Second Powers of Variable. — The last integral may be split into two parts. The first is /" \Mx = kx + C.^ which is identical with what is obtained in iig. 11. The second is I mxdx = \ma? + Cg /" which is the sweeping out of the triangular area. 65. Integral Momentum. — The following are other easy ex- amples of the first of these two formulas. The extra momentum acquired by a mass m in the interval between time t^ and time t.^^, during which its velocity is acceler- ated at the constant rate g, if its velocity be «j at time t^, is m(«2 — ''i) = I ingdt = mg{t^ - 1^ . * See Classified List, " Notation." EXAMPLES OF INTEGRATION AND DIFFERENTIATION. 33 Here mg is the acceleration of momentum, or the time-gradient of the momentum. _ 66. Integral Kinetic Energy.— The simultaneous increase of Kinetic Energy is f W - %') = f [gKh - hf + Uh - h>x} = Extra Acquired momentum x Average Velocity during interval. 67. Motion integrated for Velocity and Time. — Again the distance travelled by a train between the times t.^ and t^, when running at a constant velocity v, is to h Here the velocity v is the time-gradient of the distance travelled. 68. Motion from Acceleration and Time. — Easy examples of the second formula are the following : — If the velocity of a mass be accelerated at the uniform rate g ; then, since the velocity at any time t is {§'('- ^i) + «i}, and since in a small interval of time ht, the distance travelled is vM, where V is the average velocity during 8t, we find the distance travelled in interval {t^ - 1^ to be /: {gt - gt^ + v^}dt = -!-(«/ - V) - ^i(<2 - h) + \{h - h) ={h-h){^i+Mh-h)}- If this be multiplied by mg, we get again the increase of kinetic energy as shown above in § 66; so that the increase of kinetic energy equals the uniform acceleration of momentum (mg) multi- plied by the distance travelled.* 69. Bending Moments. — As another example, take a horizontal beam loaded uniformly with a load w per foot length. If we name by the letter I lengths along the beam from any section where we wish to find the bending moment due to this load ; then on any short length SI there is a load w.Sl, and the moment of this load upon the given section is wl.Sl, where I means the length to the • See Appendix B. 34 THE CALCULUS FOR ENGINEERS. middle of 81. The integral, or total, moment exerted upon this section by the part of the load lying between Zj and l^ is " /n k ^ = whole load on il^ — l^ multiplied by the distance of the middle of the same length from the given section. It must be noted that this is the moment exerted by the load alone independently of that exerted by the forces supporting the beam. 70. Volume of Sphere. — Passing now to volumetric integrals, we may consider a very small sectorial part of the volume of a sphere as an equal-sided cone of very smaU. vertical angle placed at the centre of the sphere, and with a very small spherical base nearly coinciding with the flat surface of small area touching the sphere. The volume of the small cone with the flat base is known to be \ the product of its base area by its height. The height here is r, the radius of the sphere. This is true whatever be the shape of the cross section of the cone. Now the whole volume of the sphere is made up of a very large number of such small-angled cones with spherical bases, these cones fitting close together so as to fill up the whole space. They would not fit close together if their cross sections were, say, circular; but the argument does not depend on the shape of the cross section, and this is to be taken such as will make the cones fit close together. In all these small conic volumes, the common factor \r appears as a constant ; each is Jj'.SA, if 8A represent the area of the small base. Thus the sum of the volumes is greater than any one of them in the same ratio as the sum of the areas of the bases is greater than the base- area of that one. Thus if A be the sum of the bases, or | dA. = A, we have the sum of the volumes equal to \rA. For any sectorial portion of the volume of the sphere, the sum of the areas of the flat tangent bases approximates to the area of the corresponding portion of the spherical surface pari passu with the approxima- tion of the sum of the flat-based conical volumes to the sum of the round-based conical volumes, which latter is the true spherical volume. Thus, if A be the area of the spherical surface, the volume subtended by it at the centre is JrA. If A be taken as the complete spherical surface, then JrA is the total spherical volume. This integration is in form identical with that of fig. 9. It differs from that in Mud, inasmuch as the differential SA is an EXAMPLES OF INTEGEATION AND DIFFERENTIATION. 35 area, while in fig. 9 the diiferential 8p is a line. The mathematical process is the same in both cases ; but the legitimacy of the appli- cation of this process depends in the one case upon the physical relations between certain curved and straight lines, while in the other case it depends on the physical relations between certain curved and flat surfaces. When it is known that the ratio of the surface-area of a sphere to the square of its radius is iir, the above integration proves the complete spherical volume to be ^^ (see § 76 below). 71. Volume of Expanding Sphere. — Consider now the spherical volume as swept through by the surface of a gradually expanding sphere. If the radius be rj at one stage of the expansion, and ?• at another, the volume swept through between these two stages is |^7r(r^ - r^^). During any small increase of size Sr from the radius (r - J8r) to (r-f J8r), the volume swept out is the normal distance Sr between the smaller and larger spherical surfaces multiplied by the mean area of the spherical surface during the motion, viz., 4irr^. That is, the increment of volume is The definite integral of this is, as above stated, and the indefinite integral for an indefinite size r is i-rr' + G. Thus 4nT^ is the r-gradient of (-f irr' + C) . If X were used to represent the radius, and X the volume, and X' the K-gradient of X and the constant factor Att be written Jc : we would here have X= lkx'dx = ^x^ + G* Expressed in words, the radius-gradient of the spherical volume is the spherical surface. 72. Volume of Expanding Pyramid. — Consider a rectangular- based pyramid of height x, and the two sides of whose base are mx and nx. The area of the base is mnx^, and, therefore, the pyram- idal volume is ^mni^. Now, suppose the size of this pyramid to be gradually increased, keeping its shape unaltered, by extending * See Classified List, III. A. 2. 36 THE CALCULUS FOR ENGINEERS. its sides in the same planes, and moving the base away from the vertex while keeping the base always parallel to its original posi- tion. As the height x increases, the sides of the rectangular base both increase in the same ratio so as to remain always ')nx and nx ; and, therefore, the increasing volume is always equal to \mnx^. As the base moves a distance 8a; away from the vertex from the height {x-\^x) to {x + \hx), the increase of volume thus added to the pyramid is the mean area of the base during this motion, viz., mva?, multiplied by the normal distance hx between the old and the new bases. The increment of volume is thus mnx^.Sx. The definite integral volume taken between the limit x^ and X2 is \mrKC^ I If the constant factor mn be written k, this result would be thus expressed, taking the indefinite form of the integral : — ikxHx = \ka? + G , which is formally or symbolically identical with the last result obtained. The difference between the two in kind is perhaps best recognised by comparing the word-expression of the last result with the following similar statement of our present one : — The height-gradient of the volume of a pyramid of given shape is the area of the base of the pyramid. In this last statement of the result no reference is made to the special shape of the cross section of the pyramid, and it is readily perceived that the reasoning employed above did not depend in any degree upon the rectangularity of the base. 73. Stress Bending Moment on Beam. — Take as another ex- ample of this formula leading from the second power in the gradient to the third power in the integral, the calculation of the stress- bending moment of a rectangular beam section exposed to pure bending of such degree as produces only stresses within the elastic limit. Under this condition the normal stress on the section in- creases uniformly with the distance from the neutral axis, which in this case is at the middle of the depth. Thus, if the whole depth of the section be called H, and the intensity of stress at the top edge (at distance — from neutral axis) be called Tc ; then the z intensity of stress at any distance h from the axis is i =^ = — ft. JH H EXAMPLES OF INTEGRATION AND DIFPEKENTIATION. 37 If the width of the section be B, the area of a small cross strip of it, of depth 8h, is B87i. If h mean the height to the middle of Sh, then the whole normal stress on this strip is -— - ■ hSh, and the moment of this round the neutral axis is 2/rE H h^ . Sh, because h is the leverage. The sum of these moments over the half of the section lying above the axis is the integral of this between the limits h = and h = JH, or ■2JfB , „ „ r2Z;B H ~3H 'T J = ^BH^ An equal sum of moments of hke sign is exerted by the stresses on the lower half of the section, and thus the Total Stress Bending Moment = -^/<;BH2.* 74. Angle Gradients of Sine and Cosine and Integration of Sine and Cosine. — In fig. 13 the angle u is supposed measured in radians, that is, in circular measure, the unit of which is the angle whose arc equals the radius. Eadians, sines, cosines, tangents, etc., are |^ pure numbers, or ratios between certain lengths and the radius of a circle; but if the radius be taken as unity, as in fig. 13, then these ratios are properly re- presented by lengths of lines, this graphic representation being to an artificial scale just as, to other artificial scales, velo- cities, moments, weights, etc., can be graphically represented by line-lengths. In fig. 13 the angle a is measured to such a scale by the length of the arc Na, while to the same scale sin a is measured Fig. 13. by as and cos a. by ac. Take a very small angle Sa, and mark off from N the two angles (a - |Sa) and (a + J8a). The horizontal and vertical projections of 8a (parallel to as and ac) are evidently the increments of the sine and cosine * See Appendix G. \ ^ ^ * 1 -- < ' 1 38 THE CALCULUS FOK ENGINEERS. for the angle increment 8a. The horizontal projection is a posi- tive increment of the sine ; the cosine decreases as a increases, so that the vertical projection is the decrement or negative increment of the cosine. If 8a be taken small enough to justify the short arc being taken as a straight line, 8a and its two projections form a small right-angled triangle of the same shape as Oas. We have, therefore, Increment of sina = 8(sina) = Horizontal projection of 8a = — x.Sa = cosa 8a aO and Decrement of cosa = - 8(cosa) = Vertical protection of Sa as » ■„ Si = — -.Su = sina oa . aO Integrating these increments between any limits Oj and oj, the results aie ""2 cosa da = and I sina cZa = I - cosa I = oosa^ - cosaj . fa-2 r n"2 sina da^l - cosa I = The student should carefully follow out this integration on the diagram through all four quadrants of the complete circle, paying attention to the changes of sign. Written as indefinite integrals these results are / cosa da = sina + C and j smada=G- cosa . * Expressed in words, this is, the angle-gradient of the sine of an angle is its cosine, and that of its cosine is its sine taken negatively. 75. Integration through 90°. — Since sin 0° = and cos 0° = 1, while sin 90° = 1 and cos 90° = 0, we find, integrating between the limits 0° and 90°, ■90° coso da=l 0° r90° I I sina da = J0° ■ and also See Classified List, VI. 1 and 2. EXAMPLES OF INTEGRATION AND DIFFERENTIATION. 39 76. Spherical Surface. — Let this result be applied to the calcu- lation of the area of the earth's surface, assuming it to be spherical. The -whole surface may be divided up into narrow rings of uniform width lying between parallels of latitude. Thus, if the difference of latitude be taken to be ^°, the uniform width of each ring will be about 17 J mUes. The meridian arc througliout this length may be considered straight without appreciable error. The ring at the equator forms practically a cylindrical ring of radius equal to that of the earth, E. A riag taken at latitude \ has a mean radius E cos \ ; and the circumferential length of its centre line is therefore 2irR cos \. Naming the difference of latitude for one ring SA., the width of the ring is E.SX, and its area therefore 2irR cos A..E.8X = 27rE^ cos A..8X. The factor QttE^ being the same for all the rings, we may first sum up all the products cos X.SX, and afterwards multiply this sum by the common factor 2irR^. If we perform this integration from the equator to the north pole, that is, between the limits X = 0° and X = 90°, we obtain the surface of the hemi- sphere. The integral of cos X.8X from 0° to 90° is 1 ; and there- fore the hemispherical surface is 2TrR^, and the whole spherical surface iirW. We used this result in § 70, p. 35. 77. Spherical Surface integrated otherwise. — The above total is 2jrE X 2E. Here 2irR is the circumference of a cylinder touch- ing the sphere, and 2E is the diameter of the sphere; so that the whole spherical surface equals that of a touching cylindrical sur- face whose length equals the dia- meter (or length) of the sphere. In fig. 14 this circumscribing cylinder is represented by its axial section nn, ss. For each strip of spherical surface of radius r bounded by parallels of latitude XX, XX, there corresponds a strip of cylindric surface U, II of radius E, which latter is, in fact, the radial projection on the cylindric surface of the spherical strip. It is easy to prove that the arc XX is greater than its projection II in the same ratio that E is greater than r. Hence the areas of the two differential strips are equal ; and, therefore, the integral areas from end to end are also equal. This proof is more elementary than that given in the previous paragraph. ^ /R---N> s Fig. 14. 40 THE CALCULUS FOR ENGINEEKS. 78. Angle-Grradients of Tangent and Co-tangent and Integra- tion of Squares of Sine and Cosine.— In fig. 15, a small angle- increment 8a is marked off equally below and above the angle o, and radii are drawn from centre through the extremities of 8a out to meet the two tangents to the quadrant of the circle at N and E. The tangent of o, or tan a, is measured along the tangent tan,(cf4Sa) ' >i* -itan a— «i from N to the radius at a, and its co-tangent, or cot a, along the tangent at E to the same radius. The increments of tan a, and of cot a, due to 8a, are marked on the figure. 8 tan u, is a positive increase of tan o for a positive increase of the angle, while 8 cot a is a decrease of cot a. The lines U and ce are drawn parallel to the short arc 8a. tt is therefore inclined to 8 tan a at the angle a, and cc to 8 cot o at the angle (90° — a). Therefore tt = cos a.8 tan o and ce= - sin a.8 cot a . Here the - sign is used in order to make co positive (8 cot a being negative). Now tt is greater than 8a in the ratio of Ot to the radius of the circle, or =-^. Similarly ee is greater than 8a in the ratio -=-^. That is, 8a = ^^.cos a = cc.cos (90° -a.) = ee sin a . Therefore, 80 = cos ^a.8 tan a = - sin ^o.S cot a . EXAMPLES OF INTEGKATION AND DIFFERENTIATION. 41 Taking all the increments minutely small, these results are written (2 tan a — ->- - =the angle-gradient of the tangent ]_ cos% and — , = the angle-gradient of the co-tangent 1 sm-'a Or otherwise / — jT-da = tan a + G J cos 'a and I -7— IT (ia = C - cot a .* J sm ^a 79. Gradient of Curve of Keciprocals. — In figure 16 there is I I 'T zwjimLm \wjMrw^^ ^ * y/' K- — X^ — ■>« ^Of-X^j- — -M U x M Fie. 16. drawn a curve of reciprocals ; the horizontal ordinate being x, the vertical ordinate is — . X See Classified List, VI. 11 and 12. 42 THE CALCULUS FOR ENGINEERS. The area of the rectangle formed by the axes and the ordinates at any point is xy. — = 1 ; constant for all points of the curve. X These two rectangles at the two points Xj^ and x overlap each other, having the common area ajj x — as part of each. Subtract this X common part and there is left xJ ) = (»•■- ^) — \os^ X j ^ " X or J_ J_ X-^ X \ This is the ratio of the decrease of — to the increase of x. When X the increments are made minutely small, — becomes practically XjX — . In the figure a small increment of x, viz., hx, is set off equally X below and above x. The above equality of areas means the equality of the two narrow strips of area rafined over in the figure. The equality is, therefore, {.-i8.}.8(l)={ 1-^8(1) }8.. fl\ ..4 <¥) Adding iSa;.8( — • ) to each side and writing instead of \x / dx changing also the sign, because — decreases while x increases, we X X dx Expressed in words this is : — The a^-gradient of the reciprocal of X is minus the reciprocal of the square of x. Writing this result inversely, we have / dx f^ 1 * X^ X * Seo Classified List, IIL A. 2. EXAMPLES OF INTEGRATION AND DIFFEKENTIATION. 43 where C is the integration constant to be determined by special limiting conditions. 80. a^Gradient of Xx and Inverse Integration. Fonnnla of Reduction. — In fig. 17 there is drawn a curve whose ordi- nates are called x and X. /' X represents any function / of a^. a; is taken to the ' middle of Bx ; and, since the | arc-length corresponding to \ 8a; is of minute length and \ may therefore be considered "'^ as straight, the point xX on the curve bisects this arc- length and also bisects SX. Also the horizontal and ver- tical lines through the point a;X on the curve divide the rectangular area ab into four equal parts, each ;|8a;.8X. The increase of the , rectangular area Xx due to the increase Sx of X is, therefore, (X -I- ^SX){x + ^Sx) - (X - |8X)(a; - JSa;) = XSa; + xSX by actual multiplication, the first and fourth terms of each product cancelling out. The first of these two terms of this increment is the strip of area between the two dotted verticals of height X ; the second is the strip between the two dotted horizontals of length x. These two strips overlap each other by the \{ab) small rectangle, and this has to be taken twice to obtain their sum. This compensates for the two strips not covering the outer small ^(ah) rectangle. Dividing by 8a;, and taking minutely small increments, that of (Xa;) being called d{Xx), and the ^-gradient of X being called X', there results ^) = X + X'a;. dx According to §§ 38 and 40, pp. 16 and 17, the X-subtangent measures X'a; ; therefore the present ai-gradient equals the sum of the function X and its X-subtangent. This X-subtangent is shown in fig. 17, where it is also graphically added to X. The result written in the inverse integration-symbolism is l(X + X'x)dx = Xsc + C. 44 THE CALCULUS FOR ENGINEEKS. As explained below in § 83, the integral j(K + X'x)dx== jXdx+ jx'xdx. Therefore the result of this article may be written jxdx = Xx- jx'xdx + C. This is an important " Eeduction Formula." * 81. a^Gradient of X/x and Inverse Integration.— In fig. 18 a curve is drawn whose ordinates are called x and X, any function of x. From two points x,X and {x + 8ic), (X + 8X) M4) A ■ 1 t i 1 ^ 1 1 X 1 1 L — 1 — ^ ^ 1 ■ir 1 X Fig. 18. -♦I&B* on this curve are drawn two straight lines to the origin 0; and on these two lines lie the upper extremities of verticals drawn at the horizontal distance 1 from 0. Evidently these last verticals measure the ratios — and -— . The difference X K + Sa; X between them is the increase of the ratio — due to the increase X 8a! of X, and is marked 8( - ) in the figure. It is less than the small height aa in the ratio of 1 to a; ; and this height aa is less than 8X by hh. This small height hb bears the same ratio to 8a; as (X + 8X) bears to (a; + hx). Thus * See Classified List, I. 8 and II. K. IMPORTANT GENERAL LAWS. 45 \x/ x\ x + Sx J Therefore, dividing by &b, dx X 3? X where, since extremely minute increments are taken, — is substi- tutedfor^Ltp.* X + 6X CHAPTEE IV. IMPORTANT QENBEAL LAWS. 82. Commutative Law. — If A-X'Sa; is to be integrated, where to each X'Sa; the same constant multiplier k is to he applied, it is evidently allowable to sum up first the series of products X'Ssc, and then to multiply this sum by k. Symbolically written this is jkX'dx^kjX'dx taken between the same limits in either case.f Keverting to the graphic representation of integration in fig. 5, the proposition means that if there he two curves drawn, of which one has at each x its height k times the other, then the first has at each X its gradient also k times as steep as that of the other. 83. Distributive Law. — If there be two curves such as in fig. 6, the height of one being called X' and that of the other H', then a third curve may be drawn, of which the height is (X' + H'). The area under the first curve is j'K'dx; that under the second is I'a'dx; that under the third is I (X' + S') ^^^ ~ ^- There is also plotted to the same scales the curve log^ x. It will be seen that the curves for TO = j'jj and re = - Jj^ lie very close together, and that the curve loge X lies between them throughout its whole length. This shows that the logarithmic curve is simply one of the general set of curves illustrating the general law, and that it is no real exception to the general law. Its position between the curves for « = ± ^^ shows that loge x is simply the special name given to the value of a;" - 1 the function when to is an excessively miuute fraction, or rather when TO is zero. Considering the variation of the curve in fig. 24 downwards from positive values of re to negative values of to, it is clear that the curve must have some definite position as n passes through zero, a position lying between that for small positive values of n and small negative values of to. This position is that PAETICULAR LAWS. 61 of the curve loe, x. For n — 0, the function takes the n indeterminate form — ^— = —^1- = - , and its value has to be found 0' by a special method, the result appearing in a special form. It should be noted that aU the curves pass through the height at the horizontal distance x=l, and that they have here one common tangent or gradient = 1. 105. Any Power of Linear Function. — If a, b, and n are constants, and X = ia + bxY, we have by § 84 and last article. Written inversely for integration, this is f{a + bxTdx = j^^ (a + &«)«+> + C * the constant C being introduced by the integration. /<' 106. Reciprocal of any Power of Linear Function. — This last integration rule fails when «= - 1. In this case we find by §§ 51 and 98, /■ 1 2"3 /(a + 6a;)-i(&!=^ log, (a + te) + C = —~ logio(a + 6a;) + C.t 107. Ratio of Two Linear Functions. — The function , """ can be reduced so as to make it depend on the last case, b + cx ah , ax a — because = <• b + cx e b + cx Therefore, by §§ 102 and 106, h ax . ax ah-, ,■, , \ , ri dx = Aog,{b + cx) + C . 'b + cx e c^ 108. Ratio of Two Linear Functions; general case. — Since the function A + Ba;_ A Bx a + bx a + bx a + bx' * See Classified List, III. A. 4. t See Classified List, III. A. 5. 62 THE CALCULUS FOR ENGINEERS. the integration of this function is performed by combining the results of §§ 106 and 107.* 109. ftuotient of Linear by Quadratic Function. — If X IX' X = a + 6a;2: then X' = 26a; and ^r-« = KT~^- ' a + bx^ 26 X Nowj^dx = [x = ^°8e X by § 98 ; therefore / dx='^log,(a + hx^) + C a+bx^ 2b where C is the integration constant. Similarly if X. = a + bx + cx^ ; then X' = 6 + 2cx, and, therefore, any function of the form A + Eg a + bx + cx^ can be readily integrated by splitting it into two terms as in § 107.t 110. Indicator Diagrams. — An important case of the use of the law of §§ J.02 and 105 is the integration of the work measured by an indicator diagram. If at any stage of the expansion p be the pressure and v be the volume of the working substance, then as the volume increases by dv, the work done is pdv. Taking the expansion law in the more general form of § 105, or p = {a + bv)'''; then the work done during expansion from p^, v^ to Pg, v^ is Expansion work done W= I pdv= j (a+hv)~''dv Here the index is always negative. If it is arithmetically greater than 1, the expansion curve makes p(a + bv) negative. But at the same time the divisor (1 — w) is negative, so that the formula makes the work done positive. It is then better to reverse the limits and to use the positive divisor (« — 1). • See Classified List, III. A. 6. t See Classified List, IIL A. 17. PARTICULAR LAWS. 63 If a = 0, or /) = &«■", as in most approximate formulas for expan- sion curves, the result simplifies, by cancelling out b from numerator and divisor, to* =^W1 W n- p, ' V- 1 --P2^i: n-\ V2 'J lb — X These formulas, which are all practically useful, give the work done during expansion in terms of the ratios between the initial and final volumes, and of the initial and final pressures ; also in terms of the initial product pv and of the final product pv. The latter formula is most useful in the case of air and gas compression pumps where the initial and known volume and pressure are v^p^. The " admission " part of the indicator diagram has an area^i«i, and this has to be added to the above, giving the total work done These calculations do not take account of the back pressure deduction from the area of the card. * The constant 6 used here equals the -n"" power of the h used in the previous formula. 64 THE CALCULUS FOR ENGINEERS. The " mean pressure " of this total area is the last value of W divided by v^, or £3 Pi \v,' w - 1 J From this the back pressure must be subtracted to obtain the " eflfective " mean pressure. In the case of isothermal gas expansion, n. = 1 oi pv= b, and the integration for work done during expansion is ■Pi Pi W = 6[log.«^ =2-3pi«i logio ^ = 2-3i>ifi logm and including the work during admission W,=m]l + 2-3 1ogio^j}- The ratio of mean to initial pressure is therefore 2^ = ^i{l + 2-31og,„^4- 111. Graphic Construction for Indicator Diagrams. — Tn fig. 25 the upper curve is a common hyperbola or curve of reciprocals, /DOC V and is the gas isothermal. The lower is drawn to the formula p = 6t)-i''i. The product pt) is the same at all points of the upper curve, and, therefore, at all points equals ^jWj. Therefore for the PARTICULAR LAWS. 65 point 2 on the lower curve, the horizontal strip of area rafined over equals (pjWj —P'f^) > ^^^ ^^i^ divided by m - 1 = '2, i.e., multiplied by 5, equals the work done under the lower curve during the expansion from 1 to 2. The mean pressure, including the admission period, therefore, equals 5 times the height of the strip rafined over plus the height to the upper edge of the same strip. The gradient of the curve p = bv''^ is negative, and equals p' = -nhv'^'^= —n£-. Therefore m = »'x — omitting the mmiis V P sign which only indicates that the forward slope is downwards. But if T be the subtangent, then p' = ^. Therefore we find V IT ra==-, and -=— -. Thus in investigating actual indicator i n— 1 w - T cards taken from engines or compressing pumps, at each point of the expansion curve at which a fair tangent can be accurately drawn, the value of the index n can be found by measuring the ratio of v to T. Also in finding the mean pressure by adding to the height of the upper edge of the rafined strip of fig. 25 the depth of this strip divided by {n - 1), this division can be per- formed very easily by an evident graphic construction, since 1 ^ T n-\ v-T Conversely, in constructing theoretical indicator diagrams, when a few points of the curve have been calculated, it much assists in the fair drawing in of the curve to draw the tangents at these points, which can easily be done by setting oif for each point T = — n If an oblique line be drawn at a tangent of inclination n to the vertical axis (it is 'drawn dotted in fig. 25)j then at each v the height of this line will give the corresponding T. In fact, by this construction the whole curve may be accurately drawn out from point to point by drawing a connected chain of short tangents whose direction is at each point obtained in this way ; the accuracy of the construction being very considerable if care be taken that each short tangent length shall stretch equally behind and in front of the point at which its direction is found by plotting T. By this construction the labour of logarithmic calculation of the heights of a series of points is rendered unnecessary.* 112. Sin-^a; and {r^ - x^)-^.—ln § 74 it was found that the angle- gradient of a sine is the cosine, and that of the cosine minus the * See Appendix E for further information concerning this class of curve, 66 THE CALCULUS FOR ENGINEBKS. sine. Tliat is, if a be the angle and s its sine, oi sin a = s ; then since cos^o =1 - s^, we have |? = (l-.2)i. da When the angle is measiwed by its sine it is symbolically ex- pressed as sin" ^s=" the angle whose sine is s." Using this nota- tion, and taking the reciprocal of the above ; i.e., taking the co- gradient or the " sine-gradient of the angle," we have d sin'^s 1 .ds (l-«2)». From this we deduce the more general result d I a sin J I ds (r^ - s*)* where a and r are constants. The corresponding integrations are, when x instead of s is used to indicate the variable, /; a dx . "'« _, ^a sin --HG = C - a cos - .* r The two angles having the same fraction for sine and cosine re- spectively are complementary ; so that these two forms of the integral only differ in the integration constants (C -G = -~-\ and in the sign of the variable parts. The sine of an angle cannot be greater than -I- 1 nor less than - 1. These integration formulse would have, therefore, no meaning in cases in which x>r or x< -r. These limits corre- spond with those within which (r^ - x^)^ remains real, because the square root of a negative quantity is " impossible " or " imaginary." If (r^ - a;^)* arises from any actual physical problem, such a prob- lem can never throughout the whole actual range of x make x>r. 113. (1 - a!^)! integrated or Area of Circular Zone. — In § 59 it was shown that the area of a sector of a circle equals Ir^a. The * See Classified List, III. B. 6 and 5. PARTICULAR LAWS. 67 angle may be expressed in terms of its sine as in last article. If s be the sine, we have Sectorial area = |r^ sin 'h. In fig. 1 3 this is the area N aO ; in which figure the length as measures rs of the present article, and ae = r cos a = r^l - s^ of the present article. The triangular area aOc, therefore, equals ^r's^l - s^. Add this to the above sectorial area ; the sum is the area ONac. This area may be taken as made up of a large number of narrow strips parallel to ON, the height of each of which would be r cos a = r^l -s^, while the horizontal width would be r.ds. The area OJSTac is, therefore, the integral of this narrow strip of area from a = o to a = a, which hmits correspond to from s = to « = s. Thus jrjl^ '^•rds = r^ Ul-sHs = ^h^jY^ + ^r^ sin -i« or Twice this is the area of a circular zone lying between a diameter and a parallel at the height s from the diameter, the radius being assumed 1 in the last equation. Here, again, s cannot range outside the limits ± 1. 114. x{r^-x')'i integrated. — The function x{r^-x^)-i may be looked on as the sine divided by the cosine, i.e., the tangent of an angle, see fig. 13, while dx is the increment of the sine. The increment of the sine multiplied by the tangent evidently equals the decrement of the cosine, and accordingly the integral is minus the cosine, or - (r^ - x^)i.i 115. (a;2 + ,.2)-i integrated. — The function {x^±r^)-i is more diflScult to deal with. Let X represent any function of x, and multiply and divide its reciprocal by (» + X) ; thus : — [ dx /■ 1 x + X^ f X^ Jx=ir^Tx'^*'=j^Tx'^^ = C + loge(a; + X)ifX' = |-. The condition X' = — — = ==- gives xdx = XdX dx X * See Classified List, III. B. 9. + See Classified List, III. B. 7. Note also that, since a; = -^—(r^-sfi) dx therefore !e{r' - !»'')~J can be recognised directly, by § 84, to be the a-gradient of -(,r'-x^)i. /■ 68 THE CALCULUS FOE ENGINEERS. or integrating a!2 + A; = X2orX = (a;2 + A;)l where the integration constant k may be either + or - . Writing 4= +r*, we have P^j = C + log,{a: + (:«'>±r')n* Here, if k is negative, the differential is " imaginary " and cannot occur in any physical problem except for values of x greater than J -k. 116. x-\r'^-x^)-^ int^rated.— The integral of a;-'(»-2 - a!2)-J is found most easily by substituting -^ for x. Thus Therefore r dx ^1 C dx _ 1 r dK = C'-hog,{x+(x^-i,y}by§115 =c-hog/-±(r!z^*.t r rx 117. Log a; integrated. — The integral of the logarithm of a variable number N is found by help of the formula of reduction in § 80 and by § 98, thus :— J log, NdN = N logj N - log6 e j" I dN + C = N{log,N-log,e} + C. logje is the "modulus" of the system of logarithms whose base is b, and for the decimal system is 0'4343 nearly. Therefore, j log.„ NrfN = N{log,„ N - -4343} + C . J * See Classified List, III. B. 6, 3, and 4. + See Classified List, III. B. 13 and 10. t See Classified List, IV. 4. PAKTICULAK LAWS. 69 118. Moment and Centre of Area of Circular Zone. — ^With the notation already used, we saw in § 113 that a narrow strip of the area of a semicircle is 2r'{l -s^^ds. The distance of this strip' from the diameter from which the angle and its sine are measured is rs, and the product of the area by this distance is the moment of the strip-area round this diameter. This is 2A(1 - s'')ids, in which r is a constant while s varies. Since s= - 4--^ , the as integral moment of all the strips for a zone between the diameter and a parallel rs away from it is easy to find. Calling (1 - s^) by letter S, we find Integral moment = 2r3 |s(l -s^)ids (SWS = -irm = |r3{l-(l-s2)«} when taken from lower limit s = or S = 1. From this is deduced by dividing by the area of the zone ; Distance of centre of area of zone from diameter ^%(l-«2)i + sin-is' For the whole half circle, sin"'s becomes a right angle or — 2 ; while s=l and (l-s^) = 0. Therefore the centre of area of a semicircle is distant from the centre of the circle by 1 4 %r.— =-=— r='4244»-. IT OTT "2 The moment of the whole semicircular area round the diameter is |/-^- The integration performed here is a geometrical illustration or proof of the general integral of x{l -x^)^.* 119. (r^ + x^y- integrated.— In § 78 it was found that the angle-gradient of the tangent equals the square of the reciprocal of the cosine. Eemembering that COS^a = ^j—— — =- 1 -1- tan^o * See Classified List, III. B. 9. 70 THE CALCULUS FOR ENGINEEKS. we find , d tan a 1 + tan^a " Call tano=<, and therefore a = i3,n'H; we then obtain the integration \da = a^\,9.n'H + C * h'h' I' Here t is essentially a number or pure ratio, and it may vary from - 00 to + 00 . If , in order to make the formula of more general apphcation, we introduce a constant r^ as follows, then t may be any + or - physical quantity, but t and r must be of the same kind. Then, since 6 with a and c both positive; then the above integra- tion of, say, (a;2 + k) may be thrown into a trigonometrical form by help of the substitution X 1 yfci X = tan"iyj- or a; = fci tan X and :5^ = — -5^ ■ «* ' X cos^X In these terms and therefore a;2 + A; = A;tan2X + fc = cos^X //(.^ + .)<£.=/.*/33l^/(^).X.* Other similar conversions of algebraic into trigonometrical in- tegrations are detailed in the II. G. Section of the Classified List. 126. Interchange of Two Fimctions. — In § 87 was estab- lished the transformation I XSdx = XH - I TUdx = XH - J H(«X a special simple case of which, already stated in § 80, is S' = 1 andB = a;, or ixdx^Xx- [x'xdx = Xx- jxdK. This general formula may be useful when the function to be integrated, viz. (XS'), is not as a whole directly integrable, but is, however, capable of being split into two factors, one of which (S') has its integral (H) directly recognisable. 127. Interchange of any number of Functions. — The opera- tion may be extended to the integration of the product of any number of functions of x according to the result of § 88 ; but with the multiplication of the number of functions to be dealt with, there is an increase in the complexity of the conditions under which the formula may be useful, and, therefore, a decrease of the probability or frequency of such usefulness. Transformations, according to this rule, are called Integration by Parts. 128. General Keduction in terms of Second Differential Coefficient.— If /(X) be any function of X, and /'(X) its X-gradient ; then, X' being the sc-gradient of X, ^J) = X'/'(X)by§84. * See Classified List, II. G. 7. 74 THE CALCULUS FOK ENGINEERS. Also "^ ~ ' (X')2 t^a; • Here is the a;-gradient of X', and is called the " second ax diflferential coeflBcient of X with respect to x," or, more simply, the "second a:-gradient of X." It is concisely written X". Using this notation (X") and applying § 126 we have jf{^)dxJ-^^j^^^,X'dx Here the given function is /'(X), and the supposition is that it is directly integrable with respect to X, but not so with respect to x. On this supposition the transformation will be of use if it is found that-^i— ^.X" is directly, or more easily, integrable with respect to X. 129. General Reduction for X'.— If /'(X) = X% then /(X) = ; 80 that in this case the above formula would be r+1 f X*" In some cases this form may be preferable to I =r,.ciX, which would be given by § 122. 130. General Keduction of ^"X''.— If in § 126 one of the two functions whose product is to be integrated be a;" and the other X', where m and r are any constant indices, the transformation gives hrx'dx=- — =^--Af hr+^X'-^dx. J m + 1 m+ 1 J If this latter quantity be not directly integrable, it may stUl be capable of being further reduced by the application of other formulas of transformation already explained, so as to finally reduce it to a directly integrable form. Such a formula is the base of certain Formulas of Reduction. 131. Conditions of Utility of Same. — The last formula given is capable of repeated apphcation, provided that X' is proportional either to some power of x or to some power of X, the right-hand TEANSFORMATIONS AND REDUCTIONS. 75 integral then reducing to the same general form as the left-hand one. In either case, or again in the case xX.' = a+ljX, it is not difficult to prove that X must be of the form X = a + te". If /• be a positive integer, then X' can be expanded into a finite series of powers of x, which when multiplied by *"' will give another series of powers of x, each term of which can be integrated separately ; so that in this case no need of the above reduction formula will arise ; although in some cases its use may shorten the work involved. But the formula is useful for repeated reductions if r is negative or fractional. Various cases of such uses are given in Section IX. of the Classi- fied Reference List at the end. 132. Reduction of x'^(a + bx% If X =a + haf, then X' = «6a!''-i=-(X'-a) and § 130 gives r 3-"'+'X'' rn C \x-^X'dx=—^- if-^ /a^-X-VX-aWa! j m+ 1 m+ 1 J ^ ' = r - — 1 Ix Xd3C+ — fx^X' Hx . m+l TO+1 J m+1 J Here we have / x^X''dx on each side. Bringing these two terms to one side, and dividing out by the sum of their numerical factors, . /, , rn \ m+l+rn VIZ. 1 -{ T I = 5 — ; ■we find \ m+lj m+l f uTXrdx^ ^"75 +-^ (afX^-'dx;* m + l+7-n m+l + rn J ' a formula of reduction by which in the integration the power of X is reduced by 1, while that of x is left unchanged. The reduction of the power of X is compensated for by the multiplication (outside the sign of integration) by the factor a, which has the same " dimen- sions " as X. This formula can be used inversely to pass from af'X''"' to jc^X"", that is, to increase the power of X by 1 without changing that of x. * See Classified List, IX. A. vi. 76 THE CALCULUS FOR ENGINEERS. If the other form of X', namely, nbx"'^, be used in this transfor- mation, there results log^YJclx = = . |a!"'+"X'-ya;: J m+l m + lj a formula of reduction by which, while the power of X is decreased by 1, that of x is increased by n. By the previous formula la;'"+"X''"^da! may be converted into a quantity in terms of 1 ai^+^X^cZa;, and thus /a!"'X'' reduced to an in- tegral in which the power of x is raised by n, while that of X is left unaltered. By similar transformations one can ring the changes among the integrals of the following set of nine functions, any one of which can be reduced to any other. afn-^r+l ^m+n^r+l i^m-n^r+l aj^X"- ^m+nX' a;™-"X'- aj^X'-' a;'»+"X'-» a-m-n-xr-l The complete set of reduction formulae for this purpose are given in Section IX. of the Classified Reference List at the end. 133. Reduction of ?■*' Power of Series of any Powers of x. — Similarly if 'S. = aa?- + ha^-\-g3Sf + k; then § 130 gives x'^'K'-dx = -- — y - ~-^ / ar{cM3e^ + phxP + ygsefyK'-'^dx . The last integral may be taken in terms each of the form laf'K''~^dx, and on account of the reduction from r to (r - 1) in the index of X, these may be more amenable to simple integration than the original \x™'KTdx. Evidently this formula applies to the sum of any number of terms in different powers of x. 134. Special Case. — If in the last article one index = and another = 1, we have X = a + 6a;" + ca; in which case a simple reduction, like that of § 132, wiU show that TRANSFORMATIONS AND REDUCTIONS. 77 (x^X'-dx = '^7^'^ + ^ — (x'^Un - l)cx + na]X'-'dx . J m+l+nr m + l+nrj '■^ ' 135. Trigonometrical Eeductions. — If in the general formula of § 126 the product XS' be equal to sin''a;, then we may split this into the two factors sin a; and sin" ''a;, thus : — X = sin"-'a: X' = (n - 1) sin""''*! cos x B' = sin a; B = - cos x X'S = - (n - 1) sin''-''a; eos'a; = - (» - 1) sin''-'^a;(l - sin '■x) = (n - 1) sia"a! - (w - 1) sia""^a; . Therefore I sin"ajc?a! = - sin""'a; cos a; - (« - 1) I wH'xdx + (n - 1 / sin""^ .L.i_ X f- IX.-W .1 1 1 4- ^JZ.. f. < « — X, -■* - JS •»! Fig. 26. 144. Integration of Second a;-gradient. — Evidently \'K."dx = X' + C. Integrating a second time {{jT'dx\dx={\ X' + clrfa: = X + CiB + K. This formula is identical with the above if C = - X'l and K = - (X^ - x^\) . This double integration of X" is written shorthand I ix"dxdx, ni or still more shortly I X^'dx^ . We thus see that when proper attention is paid to the equahty of the constants of integration, ( (d^X= I jx"dxdx or rn ru. \ dFX=l T'dx^. 145. Curvature. — To show the relation between X" and the curvature of the graphic representation, let a be the angle which the curve at any point makes with the axis of x, and let this SUCCESSrVB DIfFEEENTIATION AND MULTIPLE INTEGEATION. 83 increase to (a + Sa) in the arc-length Sa, whose horizontal projec- tion is Sx. The curvature is the reciprocal of . the radius of curva- ture, or the " arc-gradient of the angle," i.e., -r- . Since Sa is smaU, it equals tan (8a), and tan (8a) can be calculated by the ordinary trigonometrical rule from X' and (X' + 8X') the tangents of a and (a H- 8a) ; thus, 8a = tan (8a) = /^/^/^g^ ^^y = JTjrx^ nearly, when Zx is small. Also the arc-length is 8a = jW+W? = hx JT+W. Therefore the curvature is, p being the radius of curvature, 1_(^^8X' 1 _ X" p~da Zx ' {l-(-X'2}5 {l-t-X'2}!" if the radius of curvature be easily found by any direct process, the inverse form of the above relation may be useful ; namely, X" = -{l-l-X'n*. P If T be the sub tangent on the cc-axis, and if (see fig. 27) the T -»i Fig. 27. intercept on the tangent between this axis and the touching point ■y be called E ; then since W = 1L'^ + T^ and X' = ™- , therefore 84 THE CALCULUS FOR ENGINEERS. and i+x'^=5 ^(sr If the radius of curvature be known — (a practised draughtsman can always find it with the greatest accuracy in two or three seconds by one or two trials with the dividers) — the construction shown in fig. 27 affords a very easy graphic method of finding X", measiu^ed to scale, according to the above formula - ( s; ) • Those acquainted with the elements of Graphic Calculation will readily follow the construction from the marking of the figure without further explanation. 146. Harmonic Function of Sines and Cosines. — As an illustration of these ideas, take an ordinary harmonic curve. Let h be the height of the curve at horizontal ordinate I ; let rj, rj and m be constants ; and let h = r^ sin ml + j-j cos ml then h' = mr^ cos rrd - mr^ sin ml and h" = —m^ (rj sin ml + r^ cos mX) = -.m% . The student should write out these results for the three simplified oases — (1) rj — r2 = r; (2) r^^O; and (3) ri = 0. In aU cases he will find that h" = - mVi. 147. Deflection of a beam. — If a beam be uniformly loaded with a load w per ft. run and have a vertical supporting force R applied at one end, the bending moment on the section distant I from this end of the beam is M = RZ-|wZ2. The bending moment diagram (ordinates M and I) is therefore a parabola. At any point I the gradient of the curve is M' = E-«oZ = R at the end where E acts = at section where wl equals R between which points it varies uniformly. SUCCESSIVE DIFFERENTIATION AND MULTIPLE INTEGRATION. 85 The second ^-gradient of M is M"= -w and is thus constant. It is easily shown that, in a beam subjected to elastic bending M only, the curvature of the (originally straight) axis equals ==, where I is the "area-moment of inertia" of the section, and E is the modulus of elasticity. In the case of beams so stiff that the bending under safe loads is very small — which is the only case of practical interest to engineers — it is sufficiently accurate to take the curvature as the second Z-gradient of the deflection, neglecting the division by the 4 power of 1 plus the square of the first Z-gradient. Thus if A be the deflection perpendicular to Z, then the second Z-gradient of A or M 1 A " = == = ^ (RZ - ^wP) in above case and Tn , ,, ,,„ 1 [11,-r,, , ,ov „o if both E and I are A=j A" further, we measure from a line drawn through the ends of the axis so as to make the end de- flections zero, then A must be zero at Z = 0, which gives A j = 0. Inserting these values of the two constants, we find ^=m{-i^^^¥-H 86 THE CALCULUS FOR ENGINEEES. This equals zero when 1 = ; and, when ml = R, which occurs at the centre of a uniformly loaded heam freely supported at both ends, it equals B* / , , , ,\ 5 B« ^'=~3EI«8V * V 24EIw8- .If the span be L, the whole load = wL = W and R = ^wh, and 5_wIJ 5_ WL8 ^"~ 384 EI ~ 384 EI * 148. Double Integration of Sine and Cosine Function.— In § 146 we find h"=-m% and, comparing this with the original equation, we see that the general result of a double integration from this relation is A = r J sin ml + r^ cos ml where rj and r^ are the two constants introduced by integration. But if the second-gradient equation be of the other form h" = - rrfi(r^ sin ml + r^ cos ml) the result of a double integration is not the same : it is more general, namely, /i = rj sin ?w? + r^ cos ml + G-J, + Cj where Cj and C2 are the two integration constants. The former {h" = - m%) is a special case of the latter more general rule, in which special case Cj = = Cj ; and this specialty gives rise to the relation h" = - m^h, which relation does not hold good in general. In the general formula the constant C^ gives a choice of gradient of the line from which h is to be measured; while Cg gives a further choice of level at which to draw this datum line. In the special case this level must be such as to make h = r2 when Z = and the gradient of the datiun line must be zero. 149. Exponential Function.— If X = 6', then by § 95 J" X X X' = -^ = Y aid therefore X" = =2 . If X = 6"" where m is any constant, either positive or negative, whole or fractional ; then X' = 5XandX" = @X. SUCCESSIVE DIFFERENTIATION AND MULTIPLE INTEGRATION. 87 This case is the counterpart of that in the last article where t) is essentially positive. 150. Product and ftuotient of two or more a>-runctions. — If L and M are functions of x, and if X = LM, the first and second a^gradients of X are X' = L'M + LM'and .■.X" = L"M + 2L'M' + LM". In the case of M = a;, then M' = 1 and M" = ; therefore X" = L"a; + 2L'. Similarly, if X be the product of three K-functions, L, M, N, then X" is the sum of a series of terms each of which contains the three letters L M and N, and in each term the number of dashes indicating the number of differentiations will be 2. Dividing by X = LMN, the result may be written X;; L" . M" . W . JHW . L'N' . WW\ X 'l"*" M'^N "^^VLM"*" LN"*" MnJ a form analogous to that of § 92 ; and which may be extended to the product of any number of factors. 151. Third and Lower a^Gradients and Increments. — The avgradient of the second a;-gradient of X is the third a;-gradient, and the a;-gradient of this again is the fourth x-gradient ; and so on through any number of differentiations. These successive gradients are written either X'", X''', X'', or else -J^, etc. Similarly, if two successive values of the second increment of X per Sa; per hx be taken at two places 8a; apart, the difference between them is called the third increment of X per Sa; per 8a; per Sa;. This is written S'X ; and if it be divided by the cube of hi, it is S^X d^'K. easy to show that y^j = X'" = -3-5 when 8a; is very small. This is (i'X not a truism. The symbol -=-j ought not to be considered capable of being split into two parts, one of which, the numerator, (PX, is the value of 8'X when Zx is very small Nevertheless, it is correct to write for any very small 8a; 8'X = X"'(8a:)» = ^(8a;)'. 88 THE (JALOULDS FOE ENGINEERS. Again, if 8"X be the re"" increment, and -— ^ the w"" gradient, then for any very small Sx it is correct to write 152. Rational Integral a>^Functions. — If X = Itx^, then X' = Ama;"-! ; X" = km{m - l)af-^ and f^ = km(m - l)(w - 2)- - - -(w + 1 - ra)a;'"-' . If m be a positive integer, the m^ gradient of fce"" will bo a constant; ndmely, ^ = km(m-l)(m-2)..-.3-2-l=km\ Thus the (m + 1)"" gradient of kx™ is zero, as is also every lower gradient, if w be a positive integer. But if m be fractional, then the successive gradients pass into negative powers of x, so that, in this case, a lower gradient may have a very large value for very small values of x. Thus, if 5 15 15 X = x'; then, X' = y'-^^> ^" = -j-^j and X'" = -j ; giving very large values of X'" for very small values of x. If X = ocb" + bx'^~^ H + kx, and if m be an integer, then at each successive differentiation one term disappears, and the m"" gradient is again a constant, viz., am I Thus any terms in the above function, except the first, might be omitted without altering the m"" gradient. There are, therefore, (m- 1)! different functions of the above type which give the same m"' gradient ; (m - 2)! different ones which give the {m - l)"" gradient the same in all ; and so forth, the differences corresponding with those arising from putting any except the first of the constant factors in the above general formula equal to zero. 153. Lower ^-Gradient of Sine and Exponential Functions. — The successive gradients of some functions have a re-entrant or repeating character. For instance, X = l\ sin mx + k^ cos mx X"= -mFX .: X'^ = »w*X .-. X^= -m^X, etc., etc. SUCCESSIVE DIFFEEBNTIATION AND MULTIPLE INTEGRATION. 89 Again, see § 95, .•.x"=(4yx and this is true whether |8 be + or - . * 154. General Multiple Integration.— If X', X", X'", X'^ etc., are the successive K-gradients of some function X, and if we start with a knowledge of the lowest of these gradients only, and wish to work upwards to a knowledge of the higher gradients and of X by repeated integration ; we find /' where C3 is the constant of integration. Then j ix'^dx^ = lx"'dx + jc^dx = X" + C3» + Cj and and r r I c X'^(ia;3 = X' + ^x^ + G^x + C^ 'x'-c?x* = X + ^t^ + ^x^ + Cia;+ C„ . This result might perhaps be more clearly understood when ex- pressed as follows : — X" may be written (X''' + 0). Then the pro- position is that the fourth integral of the known function X'^ is the function X whose fourth a:-gradient is X", plus the function (^CgiB^ + h^.^^ + CjX + Cq) whose fourth K-gradient is 0. If there were n integrations, there would be {n + 1) terms in the result, one of which would be a constant, and (n - 1) of which would be multiples of the first (n - 1) integral' powers of x. § 152 illustrates one special example of this general proposition. The constants are to be determined from the " limiting condi- tions." The number of limiting conditions, a knowledge of which is necessary to definitely solve the problem, is the same as the num- ber of "arbitrary constants" C appearing in the general solution. * See Appendix F. 90 THE CALCULUS FOR ENGINEERS. In the above case Cj might be determined from a knowledge of one particular value of X", and then Cj from that of one particular gradient X', the remaining Cq being found from one particular value of X being given. 155. General Multiple Integration. — If in § 126 we write I Xdx instead of X, and therefore X instead of X', we obtain f {3'J Xdx}dx = 3 jxdx - jnxdx . If in this formula the a;-function be a; itself, so that E' = 1, there results ("Xdx^ = X jXdx - jxXdx which enables two possibly easy single integrations to be substituted for one double integration which may be otherwise impracticably difficult. Conversely, a given function {xX") may be difficult to integrate once, while the part of it X" is recognised as the second a;-gradient of a known function X, and then the form lxX"dx = xX'- rX"dx^ = xX'-X ■ may be useful. 156. Keduction Formulae. — From § 150 we have ^,(.X)=^X" + 2X' from which it follows that xX = rxX"dx^ + 2 ("x'dx^ . In this substitute X for X', and therefore X' for X", and I Xdx for X ; there results then rXdx^ = \x jXdx - J ("xX'dx^ = ^ I Xdx - ^ I X dX dx. SUCCESSIVE DIFFBEENTIATION AND MULTIPLE INTEGKATION. 91 Again, if in. the same formula there be substituted X for X", and therefore I Xdx for X' and / Xdx^ for X, the result appears as /m rii m Xdx^ = ^\ Xc?a;2-jl xXdx^ * 157. Graphic Diagram of Double Integration. — The meaning of double integration can be very easily represented graphically. In fig, 5 the slope of the curve is X' and the height of the curve is X, the first integral of X' by dx. Thus (XSjc) or the strip of area between two contiguous verticals under the curve is the increment of the second integral of X' by dx. Thus the area under the curve in- cluded between two given limiting verticals is their second integral, or Area under curve = / Xfda^. =P This graphic representation will help the student to perceive clearly that this integral is not the sum of a number of terms, each of which is the square of Sx multiphed by the slope X'. The square of any one Sx multiplied by the coincident slope X' would be the rectangle of base 8a; and height 8X, because X'Sx — SX.. The sum of the series of such rectangular areas stretching between given limits on the curve is not any definite area, and it can be made as small as desired by taking the 8a;'s sufficiently small. But this small rectangular area (8a;. 8X) is easily recognised to be the second increment of the area under the curve. The first diflerence is the area of the whole vertical strip between contiguous verticals. The difference between two such successive narrow strips (each being taken the same width 8a;) is the above (Sa;.8X). Thus as X.'dx^ is this second difference which equals X'(8a!)2, there is nothing illegitimate in considering the symbol dx^ in I X.'dx^ to represent the value of (Sx)^ when 8a; is taken minutely small. 1S8. Graphic Diagram of Treble Integration. — The idea of treble integration may he similarly represented graphically. If the various areas in fig. 5 under the curve measured from any given lower limit up to the various vertical ordinates at the successive values of x, be looked upon as projections or plan-sections of a solid, the successive sections for each x and the following (x + Sx) being raised above the paper by the heights x and (x + Sx) ; then mi this volume is the true graphic representation of / H'da?, because the increment of this volume, or the slice of volume lying between * See Appendix G. 92 THE CALCULUS FOR ENGINEERS. two successive parallel sections Sx apart, is the section-area at the middle of the thickness Sx multiplied by Sx. This section-area we have seen in § 157 to be I X'dx^ ; and, therefore, the above inore- ru na ment of volume is I H'da?. The integral of this is / X!d3?. If the lower limiting vertical ordinate of the area be at a! = 0, then two of the side surfaces of the above integral volume are planes normal to the paper of the diagram and passing through the axes of x and X. A third side surface, namely, that passing through the successive X edges (which are the various upper limiting ordinates in the area integrals), is also a plane : it passes through the X-axis and is inclined at 45° to the diagram paper. The fourth side surface is in general curved. These four side surfaces, three of which are flat, give to the Volumetric representation of treble integration the general form of a quadrilateral pyramid. The base of this pyramid is plane parallel to the diagram paper. As a valuable exercise, the student should endeavour to obtain a clear mental conception of the fact that X'(8a!)', the value of which becomes ~K.'doi? when hx is minutely small, is the third difference in the continuous increase with x of this pyramidal volume. CHAPTEE VIII. INDEPENDENT VARIABLES. 159. Geometrical Illustration of Two Independent Variables. — Hitherto there have been considered combinations of such functions alone as are mutually dependent on each other. The functions x, X, M, etc., have been such that no one of them can change in size without the others concurrently, changing size. In fig. 1, § 11, we have a vertical plane section of the surface of a piece of undulating land. Suppose it to be a meridional or north and south section. On it each distance measured northwards from a given starting-point corresponds to a definite elevation of the ground. If we take other meridional sections of the same piece of country, this same northward co-ordinate will correspond with other heights in these other sections. Thus, if h be used as a general symbol to mean the height of the surface at any and every point of it, then h depends not only on the northward co-ordinate INDEPENDENT VAEIABLES. 93 or latitude, but also upon the westward co-ordinate or longitude. If there be freedom to move anywhere over the surface, the two co-ordinates of latitude and longitude may be varied independently of each other, that is, a change in one does not necessitate any change in the other. Under such circumstances the elevation is said to be a function of two independent variables. 160. Equation between Independent Increments. — In moving from any point 1 to any other point 2, the elevation rises (or falls) from say h-^ to h^. Let the latitudes, or northward ordinates, of the two points be Wj and n^, an I the westward ordinates or longitudes be w^ and Wj. Then the same change of elevation would be effected by either of two pairs of motions ; namely, first, a motion northwards («2 - Wj) without change of longitude, followed by a motion (w^ - Wj) westwards without change of latitude ; or, second, a motion (to^ - Wj) without change of n, followed by a motion (v^ - «j) without change of to. This is true whether these motions be large or small. Suppose them to be small, and further suppose that there are no sudden breaks in the ground, that is, that the change of elevation is continuous or gradual over the whole surface. Call the small northward, westward, and vertical movements by the symbols TCj - »fi = 8w Wj - ?*i = Sw h2 — h^ — Sh. Then if the meridional northward slope of the ground, just north of point 1, be called [~-) , the rise during the small northward ^ J .Sn ; and if the westward slope of the parallel of latitude through 2, just east of the point 2, be ^^ I , the rise during the small westward movement Sw which, following the above, completes the motion to 2, wiU be I =r-] •S'"- The sum of these two rises gives the whole of 8/*, or \0W/2 »=©.-^-(a-^- Here the two gradients are not gradients at the same point. If Sg. 28 be a plan and two elevations of the small part of the surface 94 THE CALCULUS FOK ENGINEERS. considered, they are the northward and westward gradients at i/j and vj at the middle points of IN and N2 in the plan. If now the passage from 1 to 2 be effected by passing through W in fig. 28, and if f^\ and (^^^ be the westward and north- WesT £l.EVAriON North EL£VATI0MI PLAN Fio. 28. ward slopes of the ground at oij and m^ ; then the same change of elevation may be calculated thus, Sh- \dw/i \dnj where the Sw and the 8w are also the same lengths as before, the quadrilateral 1N2'W being a parallelogram. ( — ) and ( — ) are the westward slopes on opposite sides of this parallelogram ; they are the slopes of N2 and IW in the " North Elevation." ( — ) and ( — J are the northward slopes upon the other pair of opposite sides; they are the slopes of IN and W2 in the " West Elevation." Adding these two equations and dividing each side by 2 ; and, further, calling the means between the gradients on the opposite sides of the parallelogram by the symbols ^— and ^— ; we have dn dw' on ow INDEPENDENT VAELSiBLES. 95 On a continuous surface such as is here supposed, the above arithmetic means are, with great accuracy, equal to the actual gradients along the centre lines WjV^ and VjWg of the small rectangle ; that is, the gradients at the centre of the short straight line 1 2. 161. Equation between Independent Gradients. — If the short level length 1 2 be called Ss, so that 8n and 8w are the northward and westward projections or components of 8s ; then we have, as general truths, by dividing successively by Sn, Sw, and 8s, 8h^/dh\dh dh/dm\ Sn \dn/s dn dw'\dnjs 8h_/dh\ _^_hfdn\ dh Sw \dw)g dn\dw/, dw Sh_dk_dh /dn\ \ dh /dw\ Jb ito'\dsJs ds dn\ds where the restrictive symbol ( )b indicates a ratio of increments occurring concurrently along the special path s over the surface, an element of which path is 1 2 or 8s ; while the ratios of increments not marked with this symbol are pure northward and eastward gradients, -rr- does not need to be marked, as its terms indicate ds plainly that it means the actual whole gradient of the ground along the path s. \dw/s \dn/a \dsjs \ds /i are dififerent measures of the direction of the path s in plan ; the first two are the tangents of the inclination of this path from the west and from the north respectively ; the last two are the sines of the same inclinations. These measures of its direction par- ticularise the special path to which the equations apply. — and — dn dw have no connection with, and are quite independent of, the direc- tion of this path s : they are the due north and due west gradients at a point of the path, and depend upon the position of this point in the field, but not upon the direction of the path at such point. The gradients — and — - are called the " partial " differential on ow coefficients or gradients of h with respect to n and w. ( — ) is the ratio of rise to northward progress in travelling 96 THE CALCULUS FOE ENGINEEKS. along the path s, and depends upon the direction of this path. It is quite diiTerent from — ' . In fig. 28 it, ( — - ) , equals the tangent on \dn/s of inchnation of the line 1 2 to the horizontal base in the " West Elevation " ; while — equals the tangent of inclination of line IN on to same base also in the " West Elevation." Similarly,! 5— ) equals the tangent of inclination of 12 to the \dwjt horizontal base in the " North Elevation," while —■ is the tangent ow of inclination of line IW to same base also in the "North Elevation." 162. Constraining Relation between Three Variables. — We have above considered the ordinates n and w to any point of the surface as mutually independent of each other, and h as dependent upon ioth n and w. But we may equally well consider to a function dependent on both n and h, while looking on n and h as mutually independent of each other. Generally between the three functions n, w, and h there is only one restrictive rela- tional law established, leaving one degree of freedom of variation among the three. If a second restrictive law be imposed upon the relations between the three, this means that we are restricted to some particular path, such as s, over the surface, and are no longer free to take points all over the surface. 163. Equation of Contours. — The meridional section is such a restricted path; the restriction being 8/0 = 0. The parallel of latitude is another such restricted path ; the restriction in this case being Sra = 0. A level contour line is a third example of such a restricted path, the restriction being 8/i = 0. Therefore, if the path s be a contour line, we have -^=0, and thus one form of as the equation giving the shape of a contour is dh/dn\ dh/dw\ _ dn\ds ), 'dw\ds /, or tdn\ dh \dsjg _ /dn\ _ dw /dw\ \dw/s dh ' \ds A dn /dn\ Here {-r-j = tangent of northward bearing of contour from due INDEPEITOENT VARIABLES. 97 west, and this is seen to equal minus tlie ratio of the due west slope to the due north slope. The minus simply means that if both these slopes are positive upward gradients, then the bearing is south, not north, of due west. The steeper the west slope is in comparison with the north slope, the more does the contour veer round to the south. The geometrical linear ordinates of the above illustration may be taken as the graphic scaled representatives of any kinds of measurable quantities related to each other in a similar manner. 164. General x, y, 'E{x,y) Nomenclature. — Let the two inde- pendent variables be called x and y, and let the function dependent on these be called 'E{x,y). Let also the rate of change of F(a!,y) with x when y is kept constant be called YJ^xy), and its rate of change with y when x is kept constant be called Yy{x,y). Then the equations of § 161 are written {F',(c«,2/)}=F>,2/)| + F>,2/) where {S'J^xy)] is the rate of change of 'S{xy) with change of x when the change of x is associated with a change of y in the ratio indicated by -.^ ; this ratio -^ being any whatever, but the ratio dx dx inserted on the right hand being always the same as that involved implicitly on the left hand. 165. Two Functions of Two Independent Variables. — Again, if f{xy) be another similar function of x and y, then fdFX^l rdx,y) + Y,{x,y).% dx the brackets { } on the left meaning that the equation gives a particular value of the f{xy) - gradient of F(a;^) ; namely, that particular value obtaining along with change of y combined with change of x in the ratio -^ inserted on the right hand. dx 166. Applications to p, v, t and <^ Thermal Functions. — An important example of the kind of relation described is that of temperature, pressure, and specific volume of any one definite substance. If t, p, and v indicate these, and if H^ indicate the G 98 THE CALCULUS FOR BNGINEEES. pressure-gradient of the temperature with volume kept constant, while t\ indicates the volume-gradient of the temperature at constant pressure ; then for any changes 8p and 8w of the pressure and volume, there results a temperature increment For any change of thermal condition in which the volume-gradient of the pressure is — the volume and pressure gradients of the dv temperature are and Or again, if p', and p',, he the temperature and volume gradients of the pressure with volume and temperature respectively kept constant, then for any change defined by a volume-gradient of temperature — , the temperature and volume gradients of the pressure are and {dp\ , , dv [iy^'^'p^di _, dt \dv)=P'' dv^P- Here p'„ is the slope of the " isothermal " on the p,v diagram ; p', is the slope of the " isometric " on the p,t diagram ; t\ is the slope of the " isobaric " on the t,v diagram, etc., etc. What is called "Entropy," usually symbolised by , is most simply defined by the equation of its increment and this, combined with the above equations, gives most of the mathematical formulas of thermodynamics. 167. K-Gradient of {xy).—In fig. 17, § 80, the rectangular area between the two axes and the two co-ordinates x and X was taken as a function of these co-ordinates, and differentiated with respect to them. The problem was there considered in reference INDEPENDENT VAEIABLES. 99 to the ordinates to the particular curve shown in fig. 17, which may be regarded as similar to the particular curve s of § 161. If in fig. 17 we now regard x and X as the ordinates to any point in the whole field of the figure, they will then be independent variables. It will now be better to name the vertical ordinate y, as X is throughout this book used to indicate a function dependent on X. The area (xy) will be a function of these two independent variables. Applying the law of § 161 to this function, we have { -3^ f = -V^with y constant + 4P^with x constant x -^ { ax I ox oy dx dy , where the bracket { } indicates that the gradient is taken with concurrent change of x and y in the ratio given by y' on the right side. If the given y' be the a;-gradient of the curve drawn in fig. 17, there is here reproduced the law of § 80, which is thus shown to be simply a particular ease of a more general law, namely, that of § 161. 168. Definite Integral of runction of Independent Variables. — The equation of § 161 gives the increment of rise in level SA from any point of the surface to any other closely contiguous point. The integration of this increment of rise gives the total rise from one point to another point, near or distant, on the same surface. Taken between definite limits, this integral means the difference of level between two definite points on the surface. From any lower limit n^w^ to any upper limit ii^w^ the definite integral is(^2-/'i)- The indefinite integral is a general expression giving the height of the surface at any and every point measured from any con- venient datum level. 169. Definite Integral of Function of Independent Variables. — In integrating from point 1 to point 2 (distant from each other), the integration may be followed out along a great variety of paths, the only condition a suitable path has to fulfil being that it must pass through both points 1 and 2. The path may be curved in any fashion, or be zigzagged in any regular or irregular manner. The integration along every such path will evidently give the same result. If in fig. 28 the points 1 and 2 be distant from each other, the integration might first follow the directly north path IN, and then the directly west path N2. During the first part 8w would be continuously zero, and the integration would extend from latitude n-^ to latitude n^, keeping constantly to the longitude 100 THE CALCULUS TOR ENGINEERS. »Cj. During the second part, S« would be continuously zero. The same result is obtained by integrating first from 1 to W at constant latitude Wj, and then from W to 2 at constant longitude w^. 170. Equation between Differences of Integrals. — Incidentally it may be noted that this gives, by converting the equation between the sums of these pairs of rises into an equation between the differences of the pairs of rises on opposite sides of the rectangle, mo the left-hand expression meaning the difference between two integrations from latitude Wj to latitude Wji carried out along the meridians of longitude w^ and tOj ; while the right-hand similarly means the difference between two integrations each aloiig a parallel of latitude and each between the same Umits of longitude. 171. Indefinite Integral. — The indefinite integral h may be obtained by first integrating along any meridian up to an undefined point, and then from that same point along a parallel of latitude an indefinite distance; or the integration along the parallel of latitude may be effected first, to be followed by the meridional integration. In either case the second integration must start from the same point as that at which the first finishes, this point, however, being any whatever. 172. Independent Functional Integration Constants. — Although n and w may be varied quite independently, there is a relation between the law of the meridional section and that of the section of constant latitude which deserves notice. The equation of the meridional section, in which n is the variable, changes from section to section, i.e., changes with the longitude. This equation, therefore, in general involves the longitude w. For any one such section the value of w entering into it remains constant. Thus the general expression for h may be taken as the sum of three terms, thus : — ^. = N-l-F(w,M))-)-W where N is a function involving n but not w; W a function involving w but not n ; and F(ra,Mi) is the sum of such terms as involve both n and w. The partial gradient for any meridional section is | = N' + F>,^«) W being a constant in this differentiation. INDEPENDENT VARIABLES. ^ 101 The partial gradient for any section of equal latitude is | = W' + F>,«). These two formulEe exhibit clearly the necessary relation between the two partial gradients. They differ, first, in N' and W, which are respectively functions of n alone and of w alone, and between which parts, therefore, there is complete independence; and, secondly, in ¥\{nw) and YJjiw), which are different but not inde- pendent, being necessarily related by the condition that they are the partial gradients of the same fu^ction involving both variables. 173. Independent Functional Integration Constants. — Written in terms of independent variables x, y, and ^ the integral function of xij, these fprmulse become X = X + F(a;,y) + Y where X is a function of x only, and Y is a function of y only. 174. Complete Differentials. — In fig. 28 the slopes of the two lines IW and N2 in the " North Elevation " give the westward gradient at the two latitudes Mj and n^. These lines are drawn parallel in fig. 28 because the points 1, 2 are close together, and for a first degree of approximation the difference of slopes through them may be neglected if the surface be continuous. If a second degree of approximation to accuracy be considered ; that is, if we investigate " second gradients," the difference between these two westward gradients must be taken into account. It is evidently dn\dwj ' and the difference between the rise from N to 2 and the rise from 1 to W is d_i'dh\ dnVdioJ -j.8ra.Sw. Similarly, the difference between the northward gradients "W2 and IN as seen in the " West Elevation " of the same figure is dw\dnj ' 102 THE CALCULUS FOR ENGINEERS. and the difference between the rise from W to 2 and the rise from 1 to N is d /dh\ d_/dh\ dw\ dn) ' Sw .Sn. But by § 170 these differences equal each otlier. Cancelling out the common product 8rt.8w, we have the equality dnydwj dwKdnJ' dW Using the nomenclature of the enu of § 172, since -_— =0, because W does not involve n, and similarly — — = 0, this equation dw becomes |^(f'„0.«;)) = say ¥"„>«>) = |^(^'n(««')) = «^y ^'U^^^) ■ Thus it is indifferent whether the n or the w differentiation be taken first, and whether F"„„(«m>) or F"^(«m) be used as symbol. Although these second-gradients, calculated in these two different ways, have the same value, they represent two perfectly distinct physical phenomena. The one is the northward rate of change of the westward gradient of h. The other is the westward rate of change of the northward gradient of h. That these are equal, whatever kinds of physical quantities be represented by h, n and w, is a proposition of mathematical physics that is most interesting and fertile in its various concrete applications. 175. Second a;, //-Gradient. — When two functions of x and y fulfil the condition of being the partial x and y gradients of one and the same function, then the function formed by adding the products of these functions by Sa; and hy respectively, is said to be a complete diflferential. Thus if ^^ and -^ be the functions, ^ dx Zy ' ascertained to be the partial x and y gradients of the same function p^, then ox oy is a "complete differential," and this latter is said to be "in- tegrable." If this has been found, by accurate deduction from correct observation of physical fact, to be the increment of a real INDEPENDENT VAEIABLES. l03 physical quantity, then it is certain that the function is theoretically integrable (although the integration, may be impracticably difficult) and that its two parts will fulfil the condition of § 174. Of course, it is easy for the pure mathematician to invent functions of this sort that are not integrable, and incorrect physical observation or inaccurate deduction from physical investigation may lead to dijBferentials that are not integrable; but such have no real physical meaning. 176. Double Integration by dx and dy. — Conversely, if any function of two independent variables, x, y, be twice integrated first by dx and then by dy, the result will be the same as if first integrated by dy and then by dx, being in either case the sum of a function dependent on both x and y and of two other functions depending separately, one of them on x alone and the other on y alone. These two latter functions are introduced by the integrations in the same way as constants are introduced by integrations with respect to one variable ; the one function being a constant with respect to one variable, and the other being a constant with respect to the other variable. Thus, for example, if then , , =a + hx + ev-^exy dxdy ^- IT / h c e \ where the laws of the functions X and Y must be determined by "limiting conditions." The finding of y from the given value of -,— ^ is called the dxdy double integration of this function, and is symbolised by // T=— V- dx dy dxdy or I / 4){xy)dx dy if ^(xy) be the given functional form of — ^ J J dxdy 177. Graphic representation of Double Integration by dx and dy. — The meaning of the double integration of (l>(xy) may be represented graphically in the following different manner. Let 4>{^/) lie represented by the height of a surface from a datum plane, the co-ordinates parallel to this plane being x and y. 104 THE CALCULUS FOR ENGINEERS. Then the first integration I ^{xy)dx may be considered as extend- ing along a section perpendicular to the datum plane and parallel to the x-a.'sSs, in this integration y being a constant. The result of this integration is a general formula for the area of any such section. Two such sections at the very smaD distance 8^ apart will inclose between them, under the surface and above the datum plane, a volume equal to 8y multiplied by the area of the ^-constant section at the middle of 8j/. This volume is, therefore, I I . Sy, and the whole volume under the surface and above the datum plane therefore properly represents I I {xy)dxdy. This geometric conception is more easily grasped if the integration be taken between limits. 178. Connection between Problems concerning One Inde- pendent Variable and those concerning Two Independent Variables. — In an investigation concerning two mutually de- pendent variables, such as those in Chapters I. to VII., the two variables may always be represented by the co-ordinates to a plane curve. This curve may be looked on as a plane section of a surface, the three co-ordinates to the points upon which are related to each other by the more general kind of law dealt with in this chapter. Thus the former problems may always be conceived of as the partial solutions of more general laws connecting three variables with only one specific relation between them. The problems of Chapters I. to VII. may thus be considered special cases of more general problems of the kind now dealt with, and each of them might be deduced by specialising from a more general theorem. CHAPTEE IX. MAXIMA AND MINIMA. 179. General Criterions. — In fig. 1, at the parts C, E, H, K, R S, IT, the Z-gradient of h is zero. The points C and K are places where h rises to a maximum, the maximum K being greater than the maximum C, but the phrase " maximtim " being under- stood to mean a value greater than any neighbouring value on either side. E is a place where h falls to a minimum. MAXIMA AND MINIMA. 105 Thus the gradient falls to zero wherever there is either a maximum or a minimum value. At the maxima points, C and K, the forward gradient passes through zero hy changing from positive to negative, that is, the increase of the gradient is negative at these places. At the minimum point E, the forward gradient changes from negative to positive, so that its increase is positive. Thus the criterion for distinguishing between a maximum and a minimum is, that at the former the second gradient or second differential coefficient is negative, while at a minimum point it is positive. It is not always, however, necessary to find the sign of the second gradient in order to make sure whether the point is a maximum or a minimum. For instance, if it be known that at the place where the first gradient is zero, the value of h is positive, and if it be also known that at two points near and on either side of this place the value of h becomes zero, or of any positively less value than at this place of zero gradient, then evidently this place gives a maximum. At the place H, fig. 1, the second gradient is zero, because to the left of H it is negative, while to the right of H it is positive. This case of zero second-gradient occurring along with zero first- gradient is the limiting case coming in between the two previous ones, giving respectively maxima and minima ; and it gives neither a maximum nor a minimum. This includes the case of the dead level E,S, where also both first and second gradients are zero. Usually one's general knowledge of the physical phenomenon being investigated is sufficient, without need of evaluating the second gradient, to indicate whether or not there is any such point as H. That is, the practical man who thinks of what he is working at, and does not follow blindly mere mathematical formulas, runs substantially no risk of mistaking such a point as H for either a maximum or a minimum point. 180. Symmetry. — In very many practical problems conditions of symmetry show clearly where a maximum or minimum occurs without the need of investigating either first or second gradient. Thus, if a beam be symmetrically supported, symmetrically loaded, and have a symmetrical variation of section on either side of a certain point of its length, which point is then properly called its centre, then the bending moment and the deflection each reach a maximum at this centre. Such considerations are to be utilised wherever possible, and their use is sometimes more profitable in practical result than the more strictly mathematical process. 181. Importance of Maxima in Practical work. — As examples 106 THE CALCULUS FOK ENGINEERS. of the utility of these theorems may be cited the finding of the positions and magnitudes of maximum bending moments, of maximum stresses, of maximum deflections, of maximum velocities, of maximum accelerations of momentum, of the positions of rolling load on bridges to give maximum stress in any given member of the bridge, etc., etc. AU these things are of special importance in the practical theory of engineering. In the jointing of pieces together in machines and static structures, it is never possible to obtain uniform stress over the various important sections of the joint. It is of the greatest importance to find the maximum intensities of stress on such sections, because the safety of the construction depends on the maximum, hardly ever upon the average, stress. The average stress on the section is found by dividing the whole load on the section by the whole area of the section. Such average stresses are often very different from the maximum stress, and no reliance ought to be placed upon them as measures of strength and safety. Another class of technical problems in which maxima points are of paramount importance is that in which two or more sets of variable driving efforts, or of variable resistances, are superimposed in a machine. Thus a first approximation to the turning moment on the crank shaft of a steam engine of one cylinder, makes this moment vary as sin a, where a is the angle at which the crank stands from the dead point. If there be two cyUnders in which the total steam pressures are Pj and Pj, constant throughout the stroke, and the two cranks, keyed on the same shaft, stand apart by an angle A ; then a being the angle from dead point of one crank, (a + A) is that of the other. A remains constant while a varies. If S^ and S^ be the two strokes, the total turning moment on the shaft is ^{SiPi sina + SjPj sin (a + A)} which reaches a maximum when its a-gradient is zero; that is, when cos a _ S2P2 cos(a + A) SiPj" This ratio is minus unity when SjPj = SiPj ; and if, further, A = 90°, then a = 45° at the maximum. 182. Connecting Eod Bending Moments. — The connecting rod of an engine is at each instant bent by transverse accelerations of momentum, which, taken per inch length, would increase uniformly from zero at the crosshead to a certain amount at the crank end if the section of the rod were uniform. The MAXIMA AND MINIMA. 107 actual bending moments on the rod follow nearly the law due to this distribution of load, because the excess of weight in each head is approximately centred at the point of support at either end and, therefore, does not affect the bending moments. If L be the whole length ; I the length to any section from the crosshead ; w the transverse load per inch at the crank end : then — is the load per inch at I. On the section at I, therefore, the Li bending moment is ■! "S" ■ ^ ~ 97^ ■ "q" f • The first ^-gradient of this is zero at the point of maximum moment ; that is, this point has a distance I given by wL wP _ . or Z = i = -5773L. Inserting this value of I in the general value of the moment, we find as the maximum moment \ 6 • /3 loL L wlP 1 ■73 " L6 X 3^3 ) > = ■06415wL2 which may be compared with '0625wT?, which is the central moment in the case of the same total load, ^wL being uniformly distributed along the whole span. It is 2|% greater than this latter, and its position is 7J% of the span away from the centre. 183. Position of Supports giving Minimum Value to the Maximum Bending Moment on a Beam. — The following illustrates how maxima of arithmetic, as distinguished from algebraic, quan- tities may sometimes be found without use of a differential coefficient. If a beam, freely supported, overhang its supports equally at the two ends, and be uniformly loaded ; then certain positions for the supports will make the maximum moment less than for any other positions of these supports. Let L and w per inch be the total length and the load, and I the span between the supports. The bending moment on the section over each support is lo . -^ j- = -q- (L - 1)\ The central moment, taking it of opposite sign, is wTu I wL L _ wLf L \ "2" ■ Y ~ T" • T ~ X V ~ T/ ■ 108 THE CALCULUS FOE ENGINEERS. If this latter be negative, i.e., if i<-^, these two moments will be of the same physical sign; that is, the beam will be bent convex on its upper surface throughout its whole length. If i>-^, a certain central length will be concave on the upper surface, and inside and outside this length the moments will be of opposite sign. As Z is made larger, the magnitude of the central moment becomes always larger and that of the moment at the supports always smaller. Therefore, neither has any algebraic maximum. But when they are arithmetically equal, their common arithmetic magnitude is then less than the magnitude of the greater of the two for any other span. So that, irrespective of sign, the mini- mum of the arithmetic magnitudes of the three maximum moments is reached when or I = -SSbSL and ^^ = •2071L . Inserting this value of I in either formula for the moment we find Central moment = moment over each support = ■02144wL^ which is only 17% of the central moment on the same beam with same load when supported at the two ends. This fact may be regarded as the basis of the great economy of the modern " canti- lever " style of bridge building. 184. Position of BoUing Load for Maximum Moment and for Maximum Shear. — The next example shows how reasoning about increments, instead of differential coefficients, may be used to find maximmn values. The bending moment produced by a load on any section of a girder, supported freely at its ends, is of the same sign wherever the load be placed within the span. Therefore the moment on each and every section produced by a uniform rolling load reaches a maximum when the load covers the whole span. The right-handed integral shear stress on each section equals the supporting force at the left-hand support, minus the load apphed between this support and the section. Therefore, any load applied right of the section increases this shear stress, because it increases the left supporting force and leaves unaltered the load between it and the section. But a load applied left of the section decreases the same stress, because it increases the left supporting force less MAXIMA AND MINIMA. 109 than it increases the load between it and the section. Therefore the right-handed shear stress on any section due to a uniform rolhng load reaches a maximum when the load covers the whole of that part of the span to the right of the section, but covers none to the left of it. The left-handed shear stress reaches its maximum when the part ^eft of the section is covered. The arithmetic maximum of the stress is reached when the larger of the two segments into which the section divides the span is covered while the shorter is empty. 185. Most Economical Shape for I Girder Section. — The economic proportioning of sections is illustrated by the following. Let M be the bending moment strength of an I girder, whose depth is H outside the flanges and h inside them, and whose flange breadth is B and web thickness viE. Let the area of its cross-section be called S. Then the moment strength per square inch of section may easily be shown to be S"6H' B.~{\-w)h ■ M If H and h be increased in the same proportion, this — - will 8 increase in proportion to the first power of either of them, large sections being always stronger and stifier per square inch than small ones. It also increases if H is increased without alteration M of h. -^ also increases as w is decreased towards zero, the web S section contributing to the moment strength less than the flange section and, therefore, less than the average for the whole section. If, however, the web thickness be supposed fixed in accordance with the requirements of shear strength, and if h be diminished while H is unaltered, thereby thickening the flanges internally, this flange thickening will, up to a certain limit, increase the economy of the section, beyond which a further thickening will M decrease it again. The /i-gradient of -3- is ;;; - 3(1 - w)h^B. - (1 - w)h} + {l- w){W - (1 - w)W} 6H" {H-(l-w)/jp If this be equated to zero, there results 2(.-»,(|)'-3(i)\. = 0. This is a cubic eqjiation giving the most economical depth inside 110 THE CALCULUS FOR ENGINEERS. the flanges when that outside the flanges, as also the ratio w of web thickness to flange width, are fixed by other considerations. This ratio between h and H essentially depends on w; \iw = Q, giving zero thickness to the web, the above equation gives h = H, i.e., gives zero thickness to the flange also, or the whole section shrinks to zero area. When w= '5, it gives A/H = "6527. A useful exercise for the engineering student is to solve this equation for values of w ranging up to "5. The solution can be very easily effected by the method of solving for w taking a series of values of S./H ranging from 1 down to "6 ; tabulating these graphically as a curve ; and then reading from the curve the fe/H for any desired values of w* 186. Most Economical Proportions for a Warren Girder. — The economic proportioning of general dimensions is the subject of the next example. If a Warren girder of height H, and length of bay B, have the bay width B made up of 6 the horizontal projection of a tie-brace and (B - U) the horizontal projection of a strut brace ; then the weight of material G required to give the structure strength to carry the desired load, exclusive of that spent in jointing the various members together, may be expressed by a formula of six terms involving, besides H, B and h, also the span, the load, the stresses allowed on the sections, and four numerical coefficients which do not vary with the H nor with the span or load nor with the ratio ^ and vary very little with the number of bays. The same formula x> may be applied to any pattern of lattice girder by suitably adjusting the numerical coefficients. Four terms of this weight decrease as H increases, while two increase. A certain girder depth H will, therefore, be most economical in expenditure of material. Assuming everything but H to remain constant, and equating the H-gradient of the above to zero, there is obtained the best girder depth. Again, the girder weight contains two terms increasing with B and two others decreasing with B. Assuming the ratio -^ and all B other quantities except B to be kept unaltered and equating the B-gradient of G to zero, we obtain a formula for the most eco- nomical bay width for given span and height, which gives also indirectly the best number of bays to insert in the given span. This formula cannot, however, be precisely followed, because the number of bays must necessarily be a whole number while the equation gives in general a fractional number. The girder weight also varies with 6 in two terms, one of which * See Appendix H. MAXIMA AND MINIMA. Ill increases while the other decreases with 6. Considering every- thing but b as fixed, and putting the 6-gradient equal to zero, a rule is found for proportioning the length of the ties to that of the struts. These results are not formulated here because to guard against their incorrect application requires rather more explanation of special bridge-building detail than is suitable to this treatise. 187. Minimum Sum of Amiual Charge on Prime Cost and of Working Cost. — ^Very many technical problems are, or ought to be, solved by reducing to a minimum the sum of two main costs : first, the initial cost of construction and other necessary preliminary expenses; second, the cost of working, maintenance, and repair. These can only be added when reduced to terms rationally comparable, and this is usually done by reducing both to an annual cost or charge. Interest on all initial expenses, including prime cost of actual construction, is to be added to an annual charge to provide for a sinking fund to reproduce the capital after a period within which it is estimated that the plant will become useless from being worn out, or having become obsolete — which annual charge is often referred to as " depreciation " — and this forms the first part of the whole cost. The second part consists of wages, materials used up in working, power for driving, etc. If the initial expenditure be skiKully and wisely spent, its increase nearly always, within limits, decreases the working ex- penses. It follows that in most if not all cases a certain initial expenditure is that that will make the total annual cost a mini mum. Thus the adoption of a larger ratio of expansion in a steam engine wiU, within certain limits, diminish the consumption of water and of coal required to produce any required horse-power ; but it will necessitate a larger and more expensive engine for this same horse-power, which will be, moreover, more costly to keep in good working order ; and this is the real consideration which ought to determine the commercially most economic cut-oflf in steam engines. Lord Kelvin's calculation of the best cross-sectional area of electrical leads is another example of this kind of problem. 188. Most Economical Size for Water Pipes. — The following is a similar example directed to the calculation of the most economic diameter of water pipes, first published by the author in 1888. If a given weight or volume of water is to be delivered per hour at a certain station at a certain pressure, this means the same thing as delivering so much water horse-power at this station. Let this horse-power be called H, and the pressure demanded at the point of delivery p ; let L be the distance from the pumping or gravity- 112 THE CALCULUS FOE ENGINEEES. power station, and d be the internal diameter of the pipe. Then the loss of power in transmission, through friction and viscosity (exclusive of loss at bends and valves), can be shown to be nearly a -r™- J where a is a numerical coefficient dependent on the smooth- ness of the inside surface of the pipe and on the shape of cross section. If q be the cost per hour of generating 1 horse-power, and if the delivery be continued for T hours per year ; then the cost of this waste horse-power per year is The prime cost of pipes and pipe-laying (including trenching) may be taken as the sum of two terms, the iirst proportional to the length L, and independent of the size of pipe ; the second proportional to the quantity of metal in the pipe. The thickness of pipe requires to be designed according to the formula fA-t-^j where A and B are constants. The part of the initial cost which varies with the diameter will, therefore, give an annual cost in interest and depreciation of rLdt {--'i) where r is a factor dependent on (1) the price of iron; (2) the nature of the ground to be trenched ; and (3) the prevailing rate of interest on money. That part of the total annual cost which varies with the size of the pipe is, therefore, Equate the (X'), and suppose it expressed in terms of X'. Then from the two simultaneous equations, of which the first is the original differential equation, x=/(X') 1 and X = ^(X') + Cf X' can be eliminated so as to leave an equation involving only x and X. This is the integral solution of the given differential equation. INTEGRATION OF DIFFERENTIAL EQUATIONS. 121 As an example let X = sin X'. Then /'(X') = cosX' and f X' cos X'c^X' = X' sin X' + cos X' = u;X' + (1 - a;2)i = x sin-ia; + (1 - x^)K Therefore X = a;sin-i.B + (l-a;2)i + C. The same result is obtained by solving the given differential equation for X', viz., thus X' = sin~i2;, and integrating directly from this. 198. X=/(X'). If the implicit relation is found in the form X=/(X') either the algebraic solution of this for X' may be obtained, whence the integration /• dX _ „ or else a method similar to that of last paragraph may be followed. Thus, taking the a>gradient, whence jl^dX.' = say .^(X') = a: + C this integration by rfX' involving X' only, and giving some function of X', which is here symbolised by <^(X') . From the two simultaneous equations X=/(X')| and > x + C = (X.')\ X' is to be eliminated by ordinary algebraic means, leaving the integral equation involving only x and X. 199. mX = X'. , 1 , A particular case of the last is that of /(X') = — X'. Here 122 THE CALCULUS FOE ENGINEEKS m m ax or 1 dX m X the integration of whicli gives directly ^=llog.(CX). In fact, in this case the differential formula of § 198 reduces to that of § 195, and is the first of the two examples of the result of § 195 given in that paragraph. 200. X = :b/(X'). A differential formula only slightly different from that of § 198, and to be dealt with in the same general manner, is X = x/(X'). Taking the a-gradients of both sides, ;x:'=/(X')+^/'(X')^ from which ' and therefore /: /(X-) •_dx X' -/(X') ~ X From the two simultaneous equations X = a;/(X') and log(Ca;) = -gradients on both sides, there is obtained, X' cancelHng out from the two sides, 0={a;+/'(X')}X". This equation has two solutions. The first is X" = whence X' = CiandX = Cia; + C2 since X = a!X'+/(X') and X' = Ci. This is a partial integration of the given differential equation. The other solution is a!+/'(X') = From this and X = a;X'+/(X') treated as simultaneous equations, X' may be algebraically elimi- nated, leaving an equation giving X in terms of x either explicitly or implicitly. Let this equation be symbolised by <^(a;,X) = 0. This ^(a;,X) = is a second partial solution of the given differen- tial equation. The combination of these two partial solutions gives the complete solution in its most general form, which, in application to whatever physical problem may be in hand, must be particularised by the insertion of the " limiting conditions." These limiting conditions sometimes exclude one of the " partial " solutions as impossible, leaving the other partial solution as the full true solution of the particular physical problem in hand. The solution of a more generahsed form of this differential equation is given in § 210, the method of solution depending on that of § 208. A form differing from the last only in the sign of xX! is X-f-a;X'=/(X'). Here X + xX! is the CB-gradient of a:X. 124 THE CALCULUS FOR ENGINEERS. Therefore, the integration gives xX = jf(X.').dx = j^.dX'; and, if, X" be expressible in terms of X' alone and the function \^J be integrable by dX.', this integration will give an equation between x, X, and X', between which and the original equation, X' may be algebraically eliminated, leaving one involving only x and X. 203. Homogeneous Kational Functions. — If the relation between x, X, and X' be found in the form (ax^ + bx"'-^X + CK^-^X^ + -— )X' = Aaf + Bx^-^X + Caf-^X^ + — - where the {x,X) functions on the two sides are both "homo- geneous " of the m"^ degree, that is, where each consists of a series of products of powers of x and X, the sum of the two indices in each term being m ; then by dividing each side by a;™, this may be X converted into an equation in — . X Call — =4f) or X=x^ : therefore X' = 4p + a'-^'. X Dividing by a;"*, the diiferential equa.tion becomes (a + &J + cJ2 + ---)(J+a;J') = A + BJ + C4^2 + ---. From this is easily deduced dx _ a + hM + (iS^ + I ,«, ^ ~ A + (B-a)J + (0-6)1^2 + ■<^'9- The integral of the left side is logs;. Therefore, if that on the right is directly integrable * to a function of Si say ^( Jf )=<^f — j ; then the equation loga! + C = ^(|) gives the desired integral relation between x and X. A convenient shorthand symbol for such a homogeneous {x, X) - function of the w"" degree is f{x^^, X^). The two such functions * See Classified List, HI. A. 19. INTEGEATION OF DIFFERENTIAL EQUATIONS. 125 may be called ./(a;""'', X*") and F(a;"~'', X''). The given differential equation may then be written X'/(a:'"-'-, X') = F(a!'"-'-, X'') . Dividing /(a!™-'", X'') by a;" we obtain the same function of 1 and ^ as/(a;™~'', X'') is of x and X. The quotient may, therefore, be written /(l™-', Ml, and similarly that of F(a:'"-'-, X') by a:" may be written F(l"'-'-, ^1 ■ The integral equation then appears as r dx loga;+C= F(l"'--, J'- )_ y 204. Homogeneous Rational Functions. — The last form is a particular case of a more general one involving the first power of X' only, and the ratio only of X to x. Call this ratio M as in last article, so that as before X' = ^ + xM'. The present more general form of differential equation may be written where /( ) indicates any form of function. Therefore "^ X or dx dM X /(J)-J- The integral solution of this, namely, dM loga; + C = /(J) -J X gives X in terms of M=—, and therefore gives X also in terms of x. 205. X = k/(X'). A form of differential equation of cognate inverse character is that solved in § 200, namely, X = ^/(X') or J=/(X')whereJ = |. One solution of this is given in § 200. Otherwise, it may possibly be more easily solved algebraically so as to give X' explicitly in 126 THE CALCULUS FOR ENGINEERS. terms of M- Let f-\ ) denote the inverse of the function /( ). Then this algehraic solution would appear as of which, according to last article, the integral solution is loga; + C= I 206. X' = {Ax + BX + G)^{ax+bX + c). A differential equation bearing a Resemblance to that of § 203 is (ax + bX + e)X' = Ax + 'BX + G. If the two constants c and C did not appear, then by dividing by X, each (9;,X) function would be converted into one involving the ratio only of the two variables. But c and C can be got rid of by shifting parallelly the axes of co-ordinates from which x and X are measured, which change does not affect X'. If x and X be the new co-ordinates, then it is easily shown that the axes must be shifted so as to make Bc-bC ,^ „ Ga-cA x = x- -T-, is and. A = A. - -TT fs • A6 - aB A6 - oB Then, since — = — = X', by dividing out by x, there results an dx dx equation of the form dealt with in § 203. 207. Particular case, B = - a. If in the equation of last article B = - a, then the two terms with the common factor a combine to make the complete increment of xX. Thus the equation then reduces to (6X + c)d:X + a{Xdx -t- xdX) = {Ax + G)dx the integration of which gives ^bX^ + eX + axX-^Ax^ -0^ + ^ = in which K stands for the integration constant. 208. X'-i-X^' = S. Let ^' and S be given functions of x, of which ^' is a function whose integral by dx can be directly found, namely M. Then, if the differential equation between X, X', and x be found to be X'-hX^' = S; INTEGRATION OF DIFEEEENTIAL EQUATIONS. 127 this can be solved by the device of multiplying by what is called an " integrating factor," which means a factor which converts both sides of the equation into directly integrable functions. The factor which does this in the present case is e^, where M is the integral of the given function ^' and e is the base of the natural logarithmic system. Since the a;-gradient of e^ is e^^', that of Xe^ is X'e* + "K-X'e^- Therefore, multiplying both sides of the differential equation by e^ and integrating, there is obtained the integral equation Xe5= fHeStfe+C. This formula is of practical use only when He* is a function which can be integrated either directly or by help of some transformation. 209. X' + X^' = X''S. This process may be followed in solving the differential equation x' + x^'^X'-s because X^S is a function of x and may be inserted in place of S in the above . solution. Another solution, however, is obtained by dividing each side by X" and multiplying by the integrating factor (1 - ra)e»-"'5. The a;-gradient of XV^ is (rX' + sXJ')X'-V* so that, taking s = r, the a>_gradient of XV* is (X' + X J^')X'-' . re'*. The first factor here is identical with the left-hand side of the differential equation of this article when each side is divided by X" if ?• = 1 - ra. Therefore the integration gives Xa-ied-'ii = {l-n) I He'^""' ^dx + G. Provided He"-"'* be integrable, this formula wQl be of practical use. 210. X = a;F(X')+/(X'). The following equation is generalised from that of § 202 by inserting the general function F(X') of the a^gradient of X, in place of the special simple function X'. X = x¥{X')+f{X'). Taking the a;-gradient of each side, X' = V{X') + {x¥{X')+f'{X')}^. Transposing this, dx_ F'(X') f'jX') dX' '^ X'- F(X') X' - F(X') ■ 128 THE CALCULUS FOB ENGINEERS. This is of the same form as that of § 208 with the substitutions F(X') X'-F(X') " and /'(X') X va. place of X X „ ,, ,, a; „ 4^' X'-F(X') " " ." ^• Therefore, if we use the shorthand symbol so that 4p is a function of X' only ; and further use a for. /(X') X'-F(X')' another function of X' only ; the integration gives a;ei= fSe^dX' + C; which, if He^ be directly integrable by X', gives an algebraic equation between x and X'. Combining this with the original equation as simultaneous, the algebraic elimination of X' gives the desired integral equation involving only x and X. 211. General Equation of 1st Order of any Degree. — The pre- ceding differential equations contain X' in the first power only. The general equation of the first order and of any degree may be expressed thus : — X'» + J„_iX'»-' + X-^X'"-^ + - - - + JiX' + J^o = where 3in-\,^n-%-" ^a are n different functions involving both X and X, and n is the degree of the equation. If possible this should first be solved for X' algebraically in terms of x and X. There will be n solutions giving n values of X' which may be symbolised by X'„^ X'„_i_ X'j^ X^ each of these values of X' being expressed in terms of x and X. This reduces the above equation of the ra"" degree to an equivalent series of n linear or first-degree equations; for instance the first of this series is X'-X' =0 INTEGRATION OF DIFFERENTIAL EQUATIONS. 129 where X'„ is a function, supposed now to be known, of x and X. Integrate each such linear equation if possible, by one of the methods already given. Let the integral solutions be here symbohsed by <^„(a;, X, C) = ; <^„_i(a;, X, C) = ; etc., etc. Then the general solution, that is, the equation which includes all these various solutions, is either ^J,x, X, C) • <^„_i(^, X, C) <^lx, X, C) • U^, X, C) = or some other algebraically legitimate combination of these solutions. 212. ftuadratic Equation of First Order. — Applying the result of last article to the equation of the second degree, namely, X'2 + X'^ + H = 0. The algebraic solution of this for X' in terms of ,^ and E is 4f, 1 X'=-f +fVJ''-4S. The two integral solutions are, therefore. and X + J J^cZa; + 1 f V-^^ - 4S(^x + C = . As an easy example, take the differential equation X"i + X'sina:-^ = 0. 4 Therefore, ^^^ - 4H = ^sin^x + cos^a; = 1 ; and the two integral solu- tions are X = |-(cos x-x)-\-Qi and X = |(cos a; + a;) + C . 213. Equation of Second Order with One Variable Absent. — In differential equations of the second order with only one independent variable, there may appear powers, trigonometrical, or any other kind of functions of all the four quantities X", X', X and X, I 130 THE CALCULUS FOR ENGINEERS. Now X" may be expressed in terms of X' and either a; or X by means of the substitutions Therefore, if in any second-order differential equation X does not appear, it may be transformed by help of the substitution (a) so as to make X" also disappear, leaving only — — , X' and X . ax This is an equation of the first order between X' and x, and may be solved by methods already explained so as to give X' as a function of x (i.e., so as to eliminate -r- ) • This again is a first- order equation between X and x, and by a second similar solution we may pass to the integral equation between X and x. On the other hand, if x does not appear in any second-order equation, it may be reduced by the substitution (6), so that it will involve only ^=7 , X' and X , oX This is an equation of the first order again between X' and X, whose solution gives an equation between X' and X not involving -=-; and from this again by a second integration, the desired aX. integral relation between X and x may result.* 214. Second Order Linear Equation. — The linear, or Ist degree, equation of the second order appears in a very general form as X" + X'f{x) + X-E(x) = 4>(x), where /, F and (jt are any forms of function. Provided this equation can be solved when ^(a;) = ; then also, when (j)(x) is any function, it may be reduced to an equation of the 1st order. Thus, let B be a function of x which would be a solution if (x) were zero ; that is, let U" + n'f{x)+'BF(,x) = 0. * See Reference List, XI. C, 5f INTEGRATION OF DIFFERENTIAL EQUATIONS. 131 Give the name X to the ratio of X, the true solution of the given equation, to S ; that is, let X = JB; Then and Therefore, inserting these substitutions in the original equation, it becomes M"U + ^'{2S' + nf{x)} + M{n" + n'f(x) + mx)} = <^(a;) . But the bracketed factor of the third term is zero; so that the transformed equation becomes S'n+M'{2U'+nf{x)}=i>(x). The supposition is that H has been found ; from which H' can also be found in terms of x. This last form of the equation therefore contains only known functions of x besides X" and j^'. Now ^" is the first a^gradient of M' ; and this is therefore a 1st order linear equation as between ^' and x. Thus, if any of the already explained, or any other, method of integrating 1st order Hnear equations be applicable, then ^' can be found as an explicit or implicit function of x, thus giving another 1st order equation between M and x. By a second integration by one of these same methods, 3i may then be found as a function of x; and finally X = ^U can be obtained as the desired solution.* 215. X" + aX' + 6X = 0. The equation determining H in § 214 is soluble or not according to the particular forms of the functions /( ) and F ( ) : at any rate no reduction of the equation has yet been discovered showing, independently of the forms of /( ) and ¥{ ), how it may be solved. One simple case is that in which these functions are both constants. Let f(x)=a and 'F{x)=b, a and b being both con- stants. The equation is then, using X instead of S, X" + aX' + 6X = 0. Using the substitution (b) of § 213, this becomes '-1-6X = 0. This may be written («+rfx)^- * See Reference List, XI. C. 7. 132 THE CALCULUS FOK ENGINEERS. which is soluble by the method of § 204 ; or it may be written -^= dK' «+rfX which is soluble by § 200. By either method the solution is obtained which is printed in the Classified Reference List, at XI. C. 3. 216. X" + aX' + hX = (x). In this simple case of f{x) = a and F(x) = b, the more general equation of § 214 becomes X" + aX' + 6X = .^(a;); and its reduced form, when divided out by S, becomes *■.*•{ 4«}=^'. By § 215 both S and W are known functions of x ; and, therefore, this equation is of the form given in § 208, and can be integrated so as to give ^', provided the function •< 2-^^ + a> can be in- tegrated by dx. This is integrable because it is found that = _ a _ Jib-aHa.n | ^^ib^^+C \ when a2<46.* 217. X<'"=y(2:). If the k"' aj-gradient of X be called X'"', and the process of repeiating the integration of a function n times be symbolised by '"' . ; then, if the differential equation of the w'" order be / Xi'"=/(«) it has already been shown in § 154 that the integral equation between X and a; is fin) X = / f{x)dx^ + G„_raf-' + C„_2a!"-=' + - - - + Cia; + 0^ .t * See. Reference List, XI. 0. 6 and 3. t See Reference List, XL D 3. INTEGKATION OF DIFFERENTIAL EQUATIONS. 133 218. X<»'=/(X):X""=AX. If the equation of the w* order be X""=/(X) it is integrable only in particular cases. Thus in § 153 is given the case XW = >fcX where k is any number + or - . Let b be any number, and let Tbe the "modulus" of the system of logarithms of which the base is b. Take /8 = T/^V". Then a solution of the above equation is X = 6P'orlog6X = ;8a;. If b be taken equal to e, the base of natural logarithms, then T = 1, and the solution is log,X = 7<:V»a;. If decimal logarithms be used, or 6=10; then T = "iSi, and logi„X = -434AV''a;. Again in § 153 it is shown that if »i be an even number, and if XW = (-1)''/%X then an integral solution is X = A sin K-I'^x + B cos Ul^x where A and B are constants of integration.* 219. X"=/(X). When ra = 2, this equation becomes X"=/(X). A general rule independent of the form of /( ) has been found for iling with this second-order equation. Multiply each side by '. Then since 2X'X" is the a^gradient of X'^, and since Xldx = X'={2J/(X)rfX + A}i; from wliich, by another integration, rfX 2X' ciX, there results a; + B = {2|/(X)rfX + Ap.t * The l/ji"" root of k has n values. The insertion of these gives the iutegi'ation-ccnstants ol this »-th order equation. + See Reference List, XI. C. 5. 134 THE CALCULUS FOE ENGINEERS. As examples, the results of §§ 148 and 149 may be reproduced; but these are included in the more general formulas of last article, §218. 220. X'"'=/(Xi"-'i)- If X'"' be found as a function of X'"~'' ; then, since the a^gradient of X'"~" is X'"', if we call X'"~" by the name M, the equation may be written J'=/(J), the integration of which by dx gives -, f dM that is, give^ x'""'' as a function of x, a case which has been already dealt with in § 217.* 221. If X'"' be found as a function of X"-'' ; then, caUing X"-=' by the name ^, we have X'"' = ^", and the equation becomes the integration of which, by § 219, gives X'""''' as a function of x, and this reduces the integration to the case of § 217. t More general forms of equation, to which these last substitutions are equally applicable, are given in the Section XI. D. of the Reference Tables. 222. If /( ) and i^( ) be two functions of any form whatever, and if X=^. + f) + <^(.-f), the second gradients of X with respect to x and y may -easily be found to be ,„^ f44-?)4*"(-!)' Therefore, if the second-order differential equation rf^X , <£X^ dx"^ " dy^ be known to be true, its general integral solution is X = the above form, and the particular forms of the functions /( ) and ^( ) must be discovered from the limiting conditions of the particular concrete case. * See Reference List, XI. D. 1. t See Keference List, XI. D. 2. APPENDICES. Appendix A. — Time-Rates. (End of Chap. II., p. 28.) The Differential and Integral Calculus was first studied as an exact method of analysis of physical phenomena occurring in time, chiefly kinetic phenomena. The changes of observed physical condition occur from instant to instant, and an " instant," or small lapse of time, was taken as the common measure by which to compare simultaneously occurring changes of various kinds. Thus time was taken as the base ordinate of the diagrams which graphi- cally describe such changes. The flow of a fluid along a channel is the simplest possible illustration of such change or progress, and all phenomena were thought of as developing in the flow, or flux, of time, the universal basic increment being a small flux of time. Thus the early name given to the then new method of analysis was "Fluxions." Unless it were otherwise specified, x' or X was understood to mean the time-rate at which x increased, and the relative rates of increase of various kinds of quantities were always obtained by comparing their respective simultaneous time- rates of progress or development. So long as investigation deals with things which " take time " to develop or change in magnitude, it will be found that this original method corresponds with our innate and almost ineradicable mental habit. The corresponding increments of such things we can hardly avoid thinking of as those which are developed in the same time. Appendix B. — Enbrqy-Flux. (End of § 68, p. 33, Chap. III.) Energy manifests itself to our means of observation and measure- ment in various forms, such as kinetic, electric, thermal, luminous (light), sonorous (sound), gravity potential, electro-magnetic potential, radiant, etc. These are reciprocally convertible, and are, therefore, all measurable in like "physical dimensions," namely, MV^ or ML^T"^. As energy is, or is believed to be, indestructible, the variations it is subject to are (1) change of 136 THE CALCULUS FOR ENGINEERS. form ; (2) transference from one mass to another mass ; and (3) transference from one place to another place. The time-rate of transference of energy is horse-power ; a special unit time-rate being adopted as unit horse-power. Unfortunately, many unit time-rates of energy- variation are in use ; but they are all, of course, of the same kind, namely, horse-power. In terms of mass and velocity the measure of energy is E = ^M.V^. The time- gradient of E in a constant mass M, due to variation of velocity V in that mass, is, therefore, -;- = VM— ^ = V'F, where F is the time- at at acceleration of momentum, or the force active in the transference of energy. It may also sometimes be usefully thought of as the product of the momentum and the velocity acceleration. The space-rate or line-gradient of E with M constant is = MV— - = MV— -. -=M— =F, because V=-. dl dl dt dl dt dt Thus the two important energy-gradients are the time-gradient or horse-power, and the line-gradient or active dynamic force ; and the former equals the latter multiplied by the velocity. It is also interesting to consider the time-gradient of E with both M and V varying together. When a mass receives new energy from without, it absorbs it usually (and perhaps always) at its surface, and the new energy spreads through the mass with more or less rapidity or slowness. New impulses of kinetic energy by impact or pressure of other masses always enter and penetrate the accelerated mass in this way. In such case the time-gradient, or horse-power generating kinetic energy in the mass, is f=v..iv^_| = v(..ivf)=j|(MV,f.v*MQ|. Here — — is the time-rate at which new mass is affected by the dt , " kinetic energy, and (MV) is the whole momentum acquired at any instant. Appkndix C. — Moments op Inertia and Bending Moments. (End of § 73, p. 37, Chap. III.) The integral I bK^dh over the whole section is called the " Moment of Inertia " of the section. For an I - section with APPENDICES. 137 equal flanges and web of uniform thickness, if B be the flange width, (1-/3)B the web thickness, H the whole depth, and jjH the depth inside the flanges ; then the I = Moment of Inertia = ?5!(1 _/?^3)^ and the M = Stress Bending Moment = ft— (1 - ^r^). The sectional area is A = BH(1 -^8?;), and therefore the stress bending-moment strength per square inch of section =^=ftl ^-Pv" A 6 1-I3rj ' Girder-sections are mostly made up of rectangular parts, and the repeated application of the method here given is usually suflBcient for the calculation of their moment strength. In making such calculations, free use should be made of negative rectangular areas as parts of the section. Appendix D. — Elimination of Small Kbmainders. (End of Chap. III., p. 45.) In previous examples given, the device of taking the point a;,X at the' middle of 8a; and assuming this to correspond also to the middle of 8X, that is, assuming linear proportionality between SX and Sa;, has resulted in the exact elimination of all small X remainders. The case of — is a useful illustration of the fact X that such exact elimination does not always result from this device. In this case the result is ^^8X ^ 8X \x J , Sx ■ Sx x^ ~ iSa:^ ' x + 2 2 the small quantity ^Sa;^ not being eliminated and only disappearing "in the limit." The student should satisfy himself that the same result appears from the geometrical method followed in the text, with fig. 18 modified so as to put x and X in the centres of Sa; and SX. 138 THE CALCULUS FOR ENGINEEKS. Appendix E. — Indicator Diagrams. (End of § 111, p. 65, Chap. V.) An " indicator diagram " is any instrumental graphic record of a varying quantity. The lawjpy" = A, a constant, applies approxi- mately to very many such records when the variation is not of an elastic vibratory kind. The p and the v instrumentally observed and recorded may not be the totals measured from absolute zero of the quantities thus symbolised. For instance, p may be pressure measured from atmospheric standard, or it may be temperature v» \a _ >/* + 6) cos ^ cos ^ - sin ^ sin 6 = cos (tf> + $), give X'= p¥'^+''{A sin (px + q + O) +B cos (^« + A; + e)} X"=p^^''+'^{Asm(px + q + 2e) + Bcos{px + 7c + 20)} X""' = p''6^»:+«{ A sin (px + g + »6I) + B cos (px + k + nd)}. The only restriction among the constants is that the constant p is the same in the two harmonic functions. If x measures time, the equality of the two ^'s means that the two superimposed vibrations have equal periods or frequencies. Their difference in phase is {k — q), and this is unrestricted. In all the successive X- or time-gradients, the phase-difiference, as well as the frequency, remains the same in the two superimposed harmonics. In each gradient the frequency is the same as in the primitive. The 'phase-difference between each gradient and the next lower gradient is ^ = tan~' — i- — -. The factor J^*+» represents the damping down of the vibration, as its energy is gradually reduced by viscous or similar dissipative resistances. The amplitudes of the succes- sively higher gradients are successively less in the ratio p. This ratio p depends equally upon the frequency (the period = —) and upon the vigour of the damping coefficient b^ or ft log 6. This proposition can evidently be extended to the super- position of any number of harmonic vibrations of different amplitudes A, B, etc., and of different phases, so long as the frequency is the same in all and so long as the same damping coefficient applies to all. It is of the highest practical im- portance in modern electrical industry. See also Appendix Q, page 184. 142 the calculus foe engineers. Appendix G. — Successive Reduction Formula, (End of § 156, p. 91, Chap. VII.) The " reduction formula " of § 126 may be applied repeatedly. Such application is represented by the general formula below. The repeated application in concrete special cases is, however, simpler to appreciate than is the general result. Let X and H be any functions of x. Let D be the m'* a;-gradient of X = X""'. /(m) r nm+i) Tlien D' = X"»+>' and Idx= Udx'^K Now JBldx = D jldx - [jy I jldx I dx, Applying this repeatedly, {x'S.dx = X iudx - JX' judx^ = X [•S.dx - X' j^dx^ + jx" ^Udx^ r rm m m = XJ'Bdx-X'l ndx^ + X" Bdx^-X'" 'Bdx^+--- • r(n) r rM + x"'-"l ndx^T x'^n ndx'^+'K* By this means the function X and its derivatives are brought outside the sign of integration except in the last term. By carrying the series to the proper number («) of terms, in the last term X'"', or the w"" a;-gradient of X, may be reduced to 1 or to a constant, or to some simple function of x which combines with the »" integral of B to form an integrable function. This last term may also be transformed to Jx<"-'i I rUdx^ \ dX, in which form it may possibly be more easily integrable, * See Reference List, I. 8, w ■018 •049 •101 h XT 1 •9 •85 •8 appendices. 143 Appendix H. — Economic Proportions op I-Seotions. (End of § 185, p. 110, Chap. IX.) The economic values of — for given w, according to the equation H in the text, are as follows : — •185 •SIS ^513 ^815 1 ^ '75 -7 -65 •G -5773 In two articles in the Builders' Journal and Architectural Engineer of 15th August 1906, and in The Engineer of 9th November 1906, the author has shown that the proper propor- tioning of I-sections depends on the consideration of the maximum tensive, compressive, and shear stresses on oblique sections, which vary very differently from those on normal sections. If the web be made too thin, the maximum oblique tensive and compressive stresses at junction of web and flange are greater than at the out- side surfaces of the flanges. With proper proportioning of the web and flange thicknesses to the flange width and the total depth, the maximum oblique shear stress may be made uniform throughout the depth of the web, while at the same time the tensive and compressive stresses on oblique sections inside the flanges are made equal to those at the outside surfaces of the flanges. Using S for =-. and (1 -j8) for the above w, it is shown that H. uniformity of shear stress over the web is obtained by making 4(1-/3) _/M' 2-(l-t-/3jS2 VM H --)■ where M' and H' mean the rates at which the bending moment and the section depth vary per inch along the span, M' being, as is well known, equal to the total shear load on the section. Equality between the two most dangerous tensive and compressive stresses, at extremities of web and outside flanges, is obtained by making 16 (1-^)^ _m'j^.._ji'y (l-f-8)2(l-8) \M /■ The combination of these two equations determines a relation 144 THE CALCULUS FOE ENGINEERS. between fi and 8 independent of M, M', H, and H'. This relation is (1 + 8)2(1 - 8) = 4(1 - ^){2 - (1 + ;8)S2} , a relation which is very closely approximated to by the simpler equation ;8= -805 + 1(8 - -72)2 + (8 - -72)8. A minimum value of ^ = "805 is reached at 8 = about '7. It is shown that for the standard case of a uniformly distributed load on a beam freely supported at its two ends, these adjustments lead with very close approximation to the proportion 8= -72 ■'^' T„L^B where M^ is the central bending moment, L the span, B the flange width, and T„ the outside normal stress on flanges. Appendix I. — Economic Design op Turbines. (End of Chap. IX., p. 117.) In The Engineer of 27th May, 10th June, and 17th June 1904, will be found a series of interesting calculations by the author on "Dynamic and Commercial Economy in Turbines." It is there shown that for greatest dynamic efficiency the angles between the periphery of the rotating wheel and the blades should be equal at entrance and at exit, that the tangent of this angle should be double the tangent of the peripheral angle of the fixed entrance guide-blades, and that the peripheral velocity of the wheel should be half the peripheral component of the water entrance-velocity. The angle of the rotating blades being called ^, that of the fixed guides y, and the water-velocity through these guides w, and the wheel-velocity b, these conditions give 2 tan j8 = tan y and 6 = — cos y. The linear velocity with which the water is fed into, and dis- charged from, the wheel is then w sin y . Calling the dynamic efficiency e, the power in ft.-lbs. per APPENDICES. 145 second delivered by the water to the wheel H, and the water- covered gate-area A, we find for water-turbines € = cos^ y and H = -97 Aw^ sin y cos^ y. The smaller y is made the nearer does e approach unity ; but the y-gradient of the power developed is H' = -dlAw^ cos 7(1-3 sin2 y), and this gives maximum power at sin2y = ^ory=35° 16'with .-. ^ = 54° 44' and £=|. o This would he the best angle for the design if, for a prescribed size of wheel (measured here by A) and prescribed entrance- velocity IV, the chief aim was to obtain maximum horse-power without consideration of water-consumption. This, however, does not mean maximum commercial economy. The power " in the water " consumed is — Suppose that the e cost of each extra unit of water-power consumed is p, while the value of each extra unit of power developed (H) is h. Then the " net revenue " or " working profit " obtained from the use of the turbine is shown to be E = -97 A7m»3 sin y^^cos^ y-lV-Q where C represents initial costs incurred whatever power be taken from the wheel. The y-gradient of R is R' = -^IKhvfi cos y^l - 3 sin^ y - f ) . The maximum net revenue R for given size A and given water- velocity w is thus obtained with guide-blade angle giving sin^ 7 = if 1 - ■? I and e = _ -f- - E-, ^ ^\ hj 3 3 h' which y is less than for maximum power (H' = 0) in that S is here h subtracted from unity. Here ? is the ratio of cost of extra unit h 146 THE CALCULUS FOE ENGINEERS. of water-power consumed to value of extra unit of power utilised. With y so adjusted the maximum revenue is For different ratios of p to A the following are the results : — p h -1 -2 -3 -4 -5 -6 -7 -8 9 y degrees 35-3 33-2 3M 28-9 26-5 24-1 21-4 18-4 15-0 10'5 •373 -319 -266 -218 -173 -132 -094 -061 -033 -012 ■67 -70 -73 -76 -80 -83 -87 -90 -93 -97 As the relative cost of the water-power consumed goes up, dynamic efficiency becomes of greater commercial importance and the power extracted from the wheel with greatest commercial economy decreases. Here there is no question of varying the size of the turbine. The problem is confined to finding the best power to extract from a given size of wheel under given conditions. If the problem be how best to obtain a prescribed horse-power, the solution is quite different. The relative costs of providing a larger or smaller turbine have to be compared with the relative costs of working these at greater or smaller dynamic efficiencies. In this problem the value of the horse-power utilised is not involved. The prime cost of the installation must be reduced to a capital charge per unit of working time. The annual interest plus depreciation on the plant is divided by the number of working hours per year, and further reduced to a capital charge per second by dividing by 3600. This is taken as an initial constant not varying with the size of the turbine plus a part = aA proportional to this size as measured by the gate-area A. Adding to this the total working expenses, taken as in the other problems stated above, the total cost of the power per second is K = C-t-aA+iJ— . the constant C being different from that in the previous equations. Here H, p, and a are also constants, but A and e vary together according to the angle y chosen for the guide-blades. The larger .2A = Zh, APPENDICES. 147 this angle the less is the efficiency, but the less also is the size of turbine required to develop the prescribed H. Let K', A', and «' be the y-gradients of K, A, and t. The above equation gives K' = aA'-pH-i-. K is a minimum when K' = 0, and this gives € a which, expressed in terms of y, gives the criterion l-3sin^-y _^^,3j? l'94sin*'y a From this equation both H and A have disappeared by elimination, showing that the angle y giving maximum commercial economy does not vary with the horse-power required, and is the same for large and small sizes of turbine. In finding it the water entrance- velocity w has been assumed constant, which practically means that the available head of water is fixed. The best y depends on the cube of this velocity, and on the ratio of the extra working costs per extra unit of water-power consumed to the extra capital costs per extra square foot of gate-area in the size of turbine. The following are the numerical results of the formula : — y 10° 15° 20° 25° 30° 35° a 87 23-1 S^l 31 1-00 ■034 aAe 15-1 5-97 273 1^30 •50 •02 aA •068 •180 •415 •94 2 '67 75-4 Appendix K. — Commbrcial Economy. (End of Chap. IX., p. 117.) In all industrial applications of the doctrine of maximum and minimum values, it is most important to remember that the exact attainment of the exact maximum or minimum is never of practical importance. The method of the calculus here explained applies only to quantities with continuous variation ; and, as the 148 THE CALCULUS FOR ENGINEERS. gradient is zero at the place of maximum or minimum, on either side of this exact place there is a considerable range of base con- dition throughout which the deviation from maximum or minimum is so small as to be of no consequence. Indeed, in industrial problems the really best adjustments are hardly ever coincident with the exact values found theoretically, and for this reason that the theory never includes the consideration of every influential element. Minor elements are left out of the account in the theoretical calculation, and when the theoretical result has been found these minor elements quite rightly indicate the advisability of a small deviation from it in one or the opposite direction. In the author's work. Commercial Economy in Steam and other Heat Power- Plants, published in 1905, many very interesting physical and financial maximum problems in the economic use of steam are worked out. In this work the author enunciates for the first time a definite measure of industrial economy applicable to all productive efFort. P To this the name " economy coefiBcient " is given. It is ^=~ , 01 where P equals the value of any quantity of the product, C the cost of production of the same, and T the " time of turn over." If P be taken as the tim^-rate of production reckoned at its final value, then CT will be the " working capital " permanently held up in the maintenance of the manufacture ; so that the economy coefficient may also be expressed as " Value of Annual Production -7- Working Capital." This working capital does not include the fixed capital sunk in plant, buildings, etc. The economy thus measured is capable of being raised or lowered by changes of various kinds in the methods of production. If such change afieot all three factors P, C, and T, and if the change be capable of being made gradually, then the concurrent rates of change of these factors may be called P', C, and T', these being, say, the a;-gradients, if x be the measure of the element of manu- facture which is being varied. Maximum economy is reached, p so far as it is affected by change of x, when the !B-gradient of — CT is zero ; that is, when F^c;_ r P C T ■ This criterion of maximum commercial economy is quite general in its applicability to every kind of productive industry in its development by every sort of change capable of continuous APPENDICES. 149 gradation — whether in the manufacturing experiments the change be actually made gradually or suddenly. If P be the quantity produced per unit time and e a coefficient of physical efficiency giving the ratio of this product to the p quantity of raw material consumed, so that — is this latter £ quantity; if to be the total cost per extra unit of such raw material consumed together with the cost of working it up to the condition of the finished product ; and if p be the final value per extra unit of the product P ; then, if p and w are constant, any modification of the manufacture which affects P and « con- currently, gives a rate of variation of the net revenue -■="v{fe-')4i- Thus the maximum revenue is obtained when the rate of production is adjusted so as to make iK-P.-l. pyp w Here the size and character of plant is supposed fixed, and this equation gives the most commercially economic rate at which to work the given plant. The other most important commercial problem is to determine the best size of plant for a prescribed rate of production P. Here P is constant (P' = 0). With a larger or more expensive plant the efficiency may be raised so as to lessen the working mP expenses in proportion to — , but at the same time the capital charges are raised. These capital charges may be taken as equal to an initial constant plus hVf{i) where A is a constant factor and /(«) is a function of the efficiency dependent on the kind of industry investigated. The total annual cost for the prescribed rate of production P is thus = Constant + ftP/(e) + — , c and this is made a minimum by the adjustment w and k being among the prescribed data, and /(t) and therefore E being functions of the size or prime cost of the plant, this 150 THE CALCULUS FOR ENGINEERS. equation determines the most commercially economic size of plant to use. A particular example of the calculation has been given in Appendix I, Appendix L. — Indeterminate Forms. (End of Chap. IX., p. 117.) Whatever meaning be attached to the symbol oo , the ratio — is clearly and definitely or zero ; while the ratio — ■ is definitely 00 . But the three quantities - , — , x oo are more difficult to evaluate. They are termed " indeterminate." They arise as ratios and products of variables or fluxions, when these variables take special values, the said ratios and products having no ambiguity or indeterminateness when the variables have other than these special values. Thus X and ^ may be functions of X, both of which copie to zero for some special value of x. There is in reality no such thing as a ratio between two zeros ; a ratio can exist only between two quantities, and zero is not a quantity. The meaning attached to the symbol - must, therefore, be in a sense conventional. The meaning attached to it is the ratio of X to .Jf when these functions have any corresponding or simultaneous minutely small values. To give this meaning real significance both X and J^ must pass through zero as continuous functions of x ; therefore they can both be represented graphically by curves on a scaled diagram. Let fig. 31 illustrate such graphic representations. Both X and M curves cross the horizontal axis at the same point Xy Draw tangents to the two curves at this point. These tangents coincide with the curves for minutely small distances on either side of the touching point, and all minutely small values of X and of ^ are given equally well by the curves or by their tangents. The slopes of the tangents are X' and J^' taken at Xj, written, say, X'j and ^\. For any small + Sx on either side of a^, the values of X and J^ are thus X'jSa; and ^\Sx . Concurrent values of X and J^ are those in which the 8x is the same in both. It follows immediately that at any point close to x^ on either side of it APPENDICES. 151 As neither X'j nor ^\ is zero or ambiguous in value, ( -^ j , or - according to the meaning above assigned to this symbol, can be Fig. 31. evaluated as the ratio of the two a;-gradients at the particular lue Kj. In this demonstration it is assumed that both curves X and Jf Fig. 32. pass through the zero axis without break of gradient, that is, that both X' and Jf' are continuous. In fig. 32 are shown two pairs of curves in which, while the functions X and ^ are themselves -^ ) have the same value as ^, and can be evaluated Jp/i 152 THE CALCULUS FOR ENGINEERS. both continuous, the gradient of one of them, X', is discontinuous X' X . ■ at Xy. In this case -^„ and, therefore, also -^ , has a certain definite value for any + Sx beyond a;^ and another different definite value for any - Sx below a~. X' If -^ also assumes the form — , similar reasoning shows that by finding the ratio of these second gradients. The best graphic demonstration of this is obtained from a diagram with x as Isase and X' and Jp' as ordinates to two curves. Then X and Jp for any ± Sx on either side of ajj are the small triangular areas under the curves with common base + Sx. These areas are proportional to X' and Jf', and, therefore, also to X" and Jp". If at x^ the value of X becomes oo and that of Jp zero, then construct an a;-diagram with two curves giving = and ^. At Xi X the = curve, as also the Jp curve, both cross the zero axis, and X the above rule may serve to evaluate ( =^ ) = (Xi^), = oo x 0. \l/X/i (1 \' X' ^j = -^, we have (XJX=-(X='|;)^; from which can also be deduced (xjx=-(j2|;)^. But neither of these la.st two formulas is useful, because if X becomes oo at any finite value of x, so also does X'. The function (^j, however, may often be differentiated in terms of X so as to eliminate entirely both X and X'. Thus (XJ)i = oo X = TYh \X/i usually gives a definite value. Otherwise the product XJf may APPENDICES. 153 reduce by cancellation to a function of x which does not give an indeterminate value at x. This latter method must be adopted when X^x, because (— Y =-\ gives (a;J)„ = - a;^ J' = - oo 2 x 0, since, if ^ = at a; = 00 , necessarily 3s,' also = at same limit. To engineers the most interesting case is that of the commonly used expansion curve ju?)" = K. This gives infinite volume for zero pressure. Here«! = ( — j» and^i; = Ko' " = E"y » . In this case at zero pressure, pv = if »>1 = h „ K=l = 00 ,, n<\, the last case being for a curve lying above the hyperbolic or " gas isothermal." This last curve corresponds to expansion accompanied by very rapid heating. The work done by the expansion down to zero pressure from pjUj is (see § 110, p. 63) ■yy - Pi^i - {P«)p=o - Pi^i iin>l n—1 re- 1 = 00 „ ra )) B, IV. Logarithms and Ex- ponentials, . . .182 v. Hyperbolic Functions, . 183 VI. Trigonometric ,, 184-189 Appendix P, . . 187 ,, Q, . 188-189 VII. Inverse Functions, . 190, 191 VIIL Mixed „ 192-194 Reduction FoEMULiE, . 195-199 IX. Algebraical, Sub-section A, . . 195, 196 Trigonometrical, Sub- section B, . 197, 198 Mixed Functions, Sub- section 0, . , 199 X. Gamma Functions , 200 Diffeeential Equations, 201-207 XI. First Order, First De- gree, Sub-sec. A, 201-202 First Order, Second and Higher Degree, Sub- section B, . . . 203 Second Order, Sub-sec. C, 204 Order Higher than Se- cond, Sub-sec. D, 205, 206 Partial Differential Equa- tions, Sub-section £, . 207 APPENDICES. PASE M. Integration by Parts, . .162 Note re Inverse Use of Tables to find Differen- tial Coefficients, . . 165 N. Other Special and General Cases in Section III. . 180 PAGE 0. Extension of 7, Section IV., 182 P. Note re 25, Section VL, . 187 Q. Other Special and General Cases in Section VI., 188,180 TABLE OF CONTENTS, NOTATION, ABBREVIATIONS, GENERAL THEOREMS, . . . . . INTEGRATION BY PARTS, . . . . APPROXIMATE INTEGRATION, UNDETERMINED COEFFICIENTS, . IMAGINARY FORMS LOGARITHMIC TERMS,- . . . DIFFERENTIAL COEFFICIENTS FROM TABLES, METHODS OP IRANSFORMATION, .... RESUM]^, .... Sub-sections A to K, Detailed : — EXPANSIONS, PARTIAL FRACTIONS, SINEd AND COSINES, SUBSTITUTIONS, Sub-section D, E, F, G. Section. Pages. 159, 160 160 161-165 I. 162 163 164 164 165 165 166, 172 166, 167 II. 168 169 170 171, 172 158 CONTENTS. TABLES OP INTEGEALS, N.B. — Definite IrUegrals fmmd at end of each Section III. to VIII. ALGEBRAIC FUNCTIONS, A, Mainly Rational, ,, „ B, Quadratic Surds, APPENDIX N, A, LOGARITHMS AND EXPONENTIALS, HYPERBOLIC FUNCTIONS, TRIGONOMETRIC „ ... APPENDIX P, Q, INVERSE FUNCTIONS MIXED „ .... REDUCTION rOEMULffil, ALGEBRAICAL, . TRIGONOMETRICAL, MIXED FUNCTIONS, Sub-seotion A, B, C, GAMMA FUNCTIONS, , DIFFERENTIAL EQUATIONS FIRST ORDER, FIRST DEGREE, Sub-seotion A, „ „ SECOND AND HIGHER DEGREE, „ B, SECOND ORDER ,, C, ORDER HIGHER THAN SECOND, „ D, PARTIAL DIFFERENTIAL EQUA- TIONS, „ E, Section. Pagea. III. IV. V. VL VIL VIII. IX. XL 173-194 173-176 177-179 180 181 182 183 184-189 187 188, 189 190, 191 192-194 195-199 195, 196 197, 198 199 200 201-207 201, 202 203 204 205, 206 207 KOTATION. 159 NOTATION. Letters near the beginning of the alphabet denote/ constants which may in general be positive or negative, whole or fractional, real quantities or numbers. Those near the end of the alphabet denote variables. The symbol = stands for "denotes" or "is identical with." The symbols >, >, <, '(X)dX J ^1 j Xi where x=^{X) , . <^^(X) cZa; Sp. Case: {xdx=(^,dX 162 I. APPENDIX M — GENERAL THEOREMS, 7. 7. (Integration by Paets.) jxndx = xjndx- I ix'JHdx \dx 01 jxu'dx = xa - JTrnx = xs - [h^x Sp. Case : / Xdx = Xx - I xdX. Appendix M. — Integration by Parts: Special Cases. (I. No. 7.) The special case H' = «" is worth notice. It gives [Xx'^dx = ?^ L_ (x'x'^+^dx. J m + 1 m+lj With m = 1 : jXxdx = ^ - i- jx'x^dx. „ m = 0: jXdx = Xx— jX'xdx. „ m=-l:/ — dx=X\ogx- iX'logxdx. With X = (log a;)" : fa:'" (log xfdx = «="'^' (log «:)" _ _n Ln, /^ ^\n-ij^_ With X = loga;: fa;™loga;c?x = -^^^loga;-— J-V „ X = loga;: / H' log airfa; = B log a; - j—dx. All the formulae of Section IX. are deduced by help of this I. 7. r. GENEKAL THEOREMS, 8-10. 163 8. [ XHdx = X I H(^a; - X' T W + X" rUda^ + — - Sp. Case ■.n=l: lxdx = ^X- ^^X' + Jx" - f,X"' + etc. J 1 1'2 3! 4! 9. I ^dx = 2-3026 - - - - logi„X + C = log. X + C 10. Appboximate Intbgbation. (i.) I Xf?a;=*^(Xo+2Xi + 2X2 + — - + 2X„_, + X„) where X^ is the value of X when x = a 1> -^1 !) II J! >) — * + >• -^2 " " " " —'*"*" IJ ^r )I )> j; jj —* + ''• j: with (6 - a) divided into n equal parts giving (n + 1) values of X, (ii.) (Simpson's Kule.) "b , , Xdx='L^i^ X, + X,, + 4{X, + X, + X,+ ---.) + 2(X2 + X, + X,+ ---- + X,„_,)| where Xq , Xj , etc. = as in (i.) ; with (6 - a) divided into 2m equal parts giving (2« + 1) values of X. (iii.) f{x)dx = hh{f(b) +f{a)} +f{a + h) +f{a + 2h) ■+/(6-/i)] * See note at end of Sect, VIII, * 164 I. GENERAL THEOREMS, 11. 11. Method op Undetermined Coepfioibnts. (i.) If a function be expressible in a certain form containing unknown coefficients, these coefficients can be de- termined by transforming the identity and equating coefficients of terms whose variable part is the same function of the variable {e.g., the same power, or the same trigonometrical function), or in which the variable is absent (constant terms). The transformation referred to may be : — (a) Differentiating both sides (as in III. B. 18). (6) Clearing of fractions (as in II. E. 2, etc.). (ii.) Another method : Give to the primitive variable as many different values, in the identity, as there are coefficients to be determined ; whereby we get as many equations as are necessary to determine them. Gbnbeal Note re Imaginaries. Some of the formulas here given contain parts which would become imaginary if the quantities involved took values outside certain limits : becoming, e.g., square roots and logarithms of negative quantities, inverse sines of quantities greater than unity, etc. When a definite integral is deduced from the indefinite one, these imaginaries, explicitly or implicitly, cancel one another, if the subject of integration is itseH real. !But, in many instances, two or more forms are given for the integral of the same function (e.g.. III. B. 6, VI. 5, 6), of which that one is to be selected which, for the values of the constants in the particular problem under consideration, is free from imaginaries. Some of these formulas contain parts which are imaginary for certain values of x only, whatever the constants may be, and others do so for all values of x when the constants are outside certain limits. E.g., the formula sin"^- is imaginary when x>a, a but not when x lies between - a and + a. On the other hand, the formula — ^ sinh ~'^ { x / — \ contains imaginaries when h is Jb I V a ) negative, whatever the value of x may be. In these classified tables, the conditions under which a formula involves imaginaries are, as a rule, pointed out in cases of the latter sort, but not in those of the former. 1. GENERAL TaEOKBMS. 165 SPBCfAL Note as to Logarithmic Terms. When a term of the form A log X, where A is a constant, occurs in an integral, it becomes imaginary when x has such values as make X negative ; but in such cases A log ( - X), which is real, may always be used instead of A log X, since it has the same differential coefficient as A log X. This note applies to III. A. 3, 5, 11, 16, etc. Note re Inverse Use op Tables to find Differential COEPPICIBNTS. Although the chief purpose of these Tables is to assist in Integration, they may also be used to find gradients or differential coefficients of given functions. To use them for this purpose, search for the given function on the right-hand side of the page. Its a-gradient is the corresponding quantity on the left-hand side of the page with the sign of integration I and dx removed. The function to be differentiated will not, however, always be found under the subject-title proper to the function, since the arrangement of the tables classifies differentials, and not integrals, according to subject. For instance, the differentials of sin~'a;, ■in"', etc., sin' Algebraical.' 166 n. CHIEF METHODS OP TKANSFOKMATION, A-F. II.— CHIBr METHODS OF TRANSFORMATION. A.- Express the subject of integration as the sum of a series of terms, and integrate these separately (see I. 4). (Integration by decomposition or separation.) E.g., flog{(l + 2x){l + 3x)}dx= f {log(l + 2x) + log(l + 3x)}dx = I log(l + 2x)dx + / log(l + 3x)dx . B. Add and subtract the same quantity. E.g., [ xdx _ /' (a + 1) - I J Jl + 2x~J 1 + 2.X '^^ ^i I T~2(l + 2a;) }''''■ C. Multiply (or divide) numerator and denominator by the same quantity. . E.g., f, 3 7 _ /'tan'a;(l H-tan^a!)da; J J 1 + tan^a; _ rtanVitana: J 1 + tan^a; Sp. Case ; m + nJR m^-n^U D. Expand in a series (see p. 146). E.g., I J{1 -%mn^x) = / 1 ^ + i^"'^"'' + M^''^'" + -}dx. Sp. Case : II. F. E. Eesolve rational fractions into partial fractions. (See p. 167.) F. Express a product of powers of sines and cosines as a sum of terms, each consisting of a sine or cosine multiplied by a constant. (See p. 168.) n. CHIEB' METHODS OF TKANSFORMATION G-K. 167 G. Substitute /(X) for x and/'(X)dX for dx. (See I. 6.) Sp. Case : /(X)=i)X ■(- g = a; 1 dx = pdX / ■ In the case of a definite integral, change the limits correspond- ingly (See I. 7), or else transform back to x after integration, before assigning limits. (See p. 148.) H. Differentiate or integrate an integral with respect to any quantity in it which is not a function of x, and a new integral is deduced. K. Use integration by parts. (See I. 7.) 168 11. CHIEF METHODS OP TEA.NSFORMATION. II. D. Chief Methods of Expansion in Series : — 1 . Binomial Theorem. 2. Exponential Theorem. 3. Expansion of Log,(l ±x). 4. Trigonometric series derived from the preceding expansions, by use of imaginaries. 5. Taylor's Theorem or Maclaurin's Theorem (including 1, 2, 3, as special cases). 6. Fourier's Expansion in series of sines and cosines. 7. Spherical Harmonics, Lamp's Functions, Bessel's Functions, Toroidal Functions, etc. 11. CHIEF METHODS Or TRANSFORMATION. 169 II. E. Partial Fractions. F(a;)_Aa;"' + Ba;"-! + - - - - + H f{x) ax" + bx"-'- + + h where m and n are positive integers. Y(x) 1. If ni'^n reduce -^ by ordinary division to an integral function of z, + a fraction of similar form to the above with m Sin mx sin nx=— < cos (m - «)» - cos (m + w)a! !■ ^.ff. Sin'a; cos 2a! cos «=— sin 4a; - =-; sin 6a; - =-5 sin 2a; . 8 lo lb 2. Alternative method. i=J-\. Use Xse" .-. 2 cosa: = X + X-i 2isina; = X-X-i 2cosHa; = X'' + X-" 2i sinraa;=X"-X-". Express sines and cosines of x or its multiples in terms of X. Multiply out. Collect pairs of terms of the form C (X"±X-"), and reintroduce sines and cosines. E.g. Sin'a; cos 2a; cos x ^ (X-X-i)^(X2 + X-2)(X + X-i) (2if 22 = - 1 jt,{X« - X-« - 2(X* - X-*) + X2 - X-n 16 2« ■" — - T-R 1^^ 6a; - 2 sin 4a; + sin 2a;}. II. CHIEF METHODS OF TEANSFOEMATION. 171 II. Gr. — Substitutions. 1. {ax+bfdx==—X"dS. ] 'a I , yK^ax + b. 2. V{ax + b)dx= ~-F{X.)dX\ P Sp Case: &= ±-^. (See VI. head note.) 5 dx -dX X=i ■ xj{ax^ + b) ^{a + bXP') ^- x X=((Kt! + 6)V*. 5. 'E{ax^ + bx + c)dx = 'E{a{X? + k))d:yL v'Ka; -«)(«- 6)} 1-X2 ^v/C-^O- 7. Y{x'^ + k'^)dx = 'E{k'^s&d^X)kaw''XdX. X=tan-ii. A; Otherwise = 'E{kHosh?lL)h cosh X«?X X=sinh-i^. ft 8. r{(a;2 - A2)4}cia; = F(A; tan X)A; sec X tan XciX ft X=seo"^^-. Otherwise = F(ft sinh X)A; sinh X . rfX X=cosh~i— . 9. F{(ft2 - a;2)i}cZa; = r(ft cos X)ii cos XrfX ft" X=sin-if 172 II. CHIEF METHODS OP TRANSFORMATION. 10. r(a;, log^)dx = F(e^ X)e''dX X=log^. 11. 'E(bcosx + eamx)dx = 'F{^(b^ + c'^)smX}dK X=a; + tan"'— . c Otherwise, as in II. G. 12. 12. F(a + 5cos« + esin.)c^. = F{«-±^±2g^:iM^)^^ X=tan — . 2 1 3. F(a + 6 cos^a; + c sm^ajjtfo; = F < X2 — f 1 — X2 X=tan a; . 14. F(cosa:,sin2a;).sina;rfa;= -F{X, (1 - X2)}(iX X^cosa;. 15. F(sina;, cos^a;). cosa;(£a; = F{X, (1 - X2)}dX X=sin X , 16. F(8in-ia;)c«a; = F(X)cosXtO:, X=sin~i X and similarly for other inverse functions. TABLE OF INTEGRALS, IIL A. 1-10. 173 III.-IX.-TABLE OF INTEGRALS. III. ALGEBRAIC FUNCTIONS. III. A. Mainly Rational. 1. \adx =C + ax. 2. CBMa; =C+? . / « + l [Exc. w = - 1. (See III. A. 3.)] 3. [^ =0 + 2-302585---- xlogios; = 0-1- log^a; . 4. {(ax \ hfdx = C + , V («•« + *)''+^ j^ (ra-t-l)a^ ' Exc. n=-\. (See III. A. 5.) - f dx ri , 2'3026 , / , ,\ 5. I =^ = + X logUax + o) J ax + b a = C + —log,{ax + i). 6. I =~rfa; = C-)- — x+ = — logAax + b). J ax + b a (f ' 7. lar{ax + bfdx. Use II. A., or IX. A. 1, or III. A. 20. 8. f-^„ =C-Htan-'a! = C-cot-ia;, 9. f-^ =C-t-|log,l±? = C-f-tanh-'a; [x 1] . 10. f -— = -7^^. tan-'xj %+C when a6 > . (See III. A. 9.) 174 in. ALGEBRAIC FUNCTIONS, A. 11-15. ,, f dx __!__, / Aa: + B \ ■ j{Ax + B)(ax + b)~^'^Ab-aB°^'\ax + bJ' 12. \- J where to is a positive iateger or 0, and n a positive J l+af integer, li m<^n, use II. E. 1 ; it m log,(^^ - 2x cos r^^ + 1)} 2 -^ ) . r(m+l)7r, .1/ H 2,-{ sm-i ^tan M Ix - cos — \ n \ N.B. — If n is odd, r takes the values 1, 3, 5 (w - 2). If TC is even, r „ „ 1, 3, 5 {n-Vj, and the term i ^Ioge(l +a;) is omitted. f x^dx 1 3. I — ^ where wi is a positive integer or 0, and n is an odd J ^ -^ positive integer. = ( - l)™+i/'?!^ , where X= - x. (See III. A. 12.) fx^dx 14. I ^ where wi is a positive integer or 0, and n is an even ■^ positive integer. = c + liogXi -x)- i::^ iog,(i + x) _ ly i co/('^ + l)" log,(:.^ _ 2x 00^"^+ 1) I n ■ r(m+l)ir, J sm-^^ ^tan ^ sm — n where r takes the values 2, 4, 6 (w - 2). See III. A. 12, a m and w are positive integers. = t( - W i r^ "^^"-^ ^K -t) ''' % _ See III. A. 13, 14, if m and n are positive integers. Otherwise, see IX. A. 1. m. ALGEBRAIC FUNCTIONS, A. 16-20. 175 16. f-^l =C + ^A^^tan->^^+A where A^V-iae, Jax' + bx + c V(-A) V(-A) ifA<0 = 0+ 1 log.|^^±*ZL^ ifA>0. , „ f(Ax + B)dx A , / 2 , 7, , \ , 2aB - Ah f dx 17. 1^—5 — =-^i — =—-\oQ.{ax^ + ox + c) + I — s — i • jax' + lx + c 2a ^'^ '^ 2a J ace" + 6a; + c (See III. A. 16.) 18. I ^= — dx where X=aa:^ + bx + c J X" = ;w , , ^^ — r-\ 1=^. (See IX. A. 4.) 2(?i-l)aX»-i 2a ;X" ^ ^ Sp. Case : A = 0. (See IX. A. 4.) fAx™ + Jix'"~^4- - - + K 19. — :r—- —- dx, where m, n are positive integers -- J ax" + bx''-^ + +p ' ' f 6 Keduce by II. E. to terms like these : — JAx^dx, 1^^^, 1^^, I ,^-,+ ^ dx, Jr^±^J^; J Jx-p J{x-pY Jx^ + lx + m J{x' + lx + my for which, see III. A. 2, 5, 4, 17, 18. 20. \x^{ax'' + hyi<^dx = S^\'X?*''-'-{± — °\ /S-IX^+9-lc?X where X=(6a;-» + a)W«. Use the former when is a positive integer. n latter „ -—- + ^ is a negative integer. " n q In either case, expand the binomial factor and use II. A. and III. A. 2. 176 III. ALGEBRAIC FUNCTIONS, A. 21-27. Definite Integrals with Numerical {or Particular) Limits. = ; ^ where n and m are even positive in- 0^ + ^" n sin <"' + ^)^ tegers, and m < « . n 2 I ^^ ■J 0^(1- on I ax IT IT .» 1 22. I — = — cosec — iira>l, ' -'" -af) M n 23. I ,/=" , =-!Lcot^ „ „ „;/(l+a!") n n " " „. I yjo -TJo ]u,Ji _i.\im,-vV)V{n-irV) (See X. 1-6.) 25. 26. r{x'"+x^)dx _ r(m+i)r(w- I (1+^)'"+"+^ ~ r(m + M + i I ^ ' r(m + w + 2) J ^ -a;)™+' i} II )» ») 27. rV(a_c«)Ma; = a™+»+^%±lM^)» I ^ ' r(m + « + 2) III. ALGEBRAIC FUNCTIONS, B. 1-7. 177 III. B. Quadratic Surds. 1. \{ax + 'b)idx = G + ^{ax + V)i. 2. J/Km + hf}dx = ? jXf{X)d{X) where X=(aa; + l>)i . Sp. Cases : — (-?) J x{ax + V)Mx = C + ^{ax + &)l - ~lax + 5)? . («) [ -7-^i = 2 f ^ . (See III. A. 10.) 3- /(^^j =C + log,{a; + (a=^+l)n=C + sinh-^. 4. [ .f'" =C + loge{a; + (a^-l)i}=C + cosh-ia!. 5- / 7^ s^T = C + sin"ia; = C - cos~ia; . J (I- ar)i 6. f , f'^.vx -C + i-log,{a!> + (aa;^ + S)J} if a>0and6>0 = C + -^sinh-iwA/r „ a>0 „ 6>0 = + -^ cosh-la; ^Jl? „fl>o „ 6<0 = C + ;^)Sin-V-f „«<0 „ 6>0 Otherwise : put x=X /- or a;= X . /( — \ and use III. B. Va VV «; 3, 4, or 5. 7. f.-?^ =C+i(aa;'^ + &)». 178 III. ALGEBRAIC FUNCTIONS, B. 8-16. 8. /"__^__ - _ f-J^ where X=^ . (See III. B. 6.) Sp. Case : / — sn = C - sinh~' - = C - cosech"' x Jx{l+x^y X I — — r = C - cosh"' _ = C - sech"* x . jx{i-x^y X 9. I {Ax + -B\ax' + b)hdx = C + {^x^^ + |a; + ^)(«a:' + *)* + ?fr^AU- (See III. B. 6.) 2 J {ax' + o)i 8p. Case : L{ax'' + h)Hx = C + ^{ax^ + bf. 1 0. I ,-77 3ri = C + vers-i- = C + cos-^ . J {2ax - arp a a ■in [ d^ _p (^ax±si?)^ J x{2ax±x')i~ ax ,„ /■ dx n , f ^X ■ J (ax^ + 6a! + c)4 '^ ^'*j {ia^T^ + 4ac - 62)» * Where X^a; + A . (See III. B. 6.) , , /■ (A a; + E)da! _ A , , . , 2Ba - A& /'___if^___ j(aa;'^ + te + c)i"a^'*^"^'* + ''^ "^ 2a J {ax'+bx + e)i' (See II. B. 13.) 15. f(Aa; + B)(aa?+6a! + c)i£?a; p^jA, /B A6\ B& Ac A6^K »,, , ., /Bc_A6c_B6^ A&'W rfa; \ 2 4a 8a "•" 16aVi (aa;^ + 69; + c)* * (See III. B. 13.) ,g f (Aa; + B)f?a; ^ ^ 6E 2eA + (2aB - 6A)ie ■ j (aa;2 + bx + c)i~^'^ "^ (4ac - J^)(aa:» + Sa; + c)* " III. ALGEBRAIC FUNCTIONS, B. 17-18. 179 J {ax' + bx + c)'^i «a'-^+ {ax' + bx + ef-i if w is a positive integer ; where L, M K. are constants to be determined by I. 11 (a). J (ax^ + bx + c)i Mdx = (Pk"-' + qx'^-' + ---- + S)(aa;2 + bx + e)i + f^-^ y (aaf + (aa;^ + fta; + c)5 where the constants P, Q, S, M are determined by I. 11. For the last integral, see III. B. 13. Otherwise, see IX. A. 3. 180 III. ALGEBRAIC FUNCTIONS — APPENDIX N. Appendix N. — Section III. Some Special and some MORE General Cases. A (4). Since ^=1--^, .". f-_^ = C + ^-^. ^ ' x + b z + b' }{x-vhf b x + b A (8). f_^=C + -^tan-'fx^/^). ^ ' jax'^^-b ^ \ V b/ A (9). f^,^=C + llog^±f'. A (12). "When n=2, m may be or 1 ; and the formula gives ■ When n = 3,m may be , or 1 , or 2 ; and the formula gives with m = 0,jj-j^g= -Iog(l+a;)- g log(a;2-a!+ 1) 4-577tan-^ + C, and with OT= 1,1 r^-A= - ^log{l+x)+ - log (j;^ - a; + 1) 4-577 tan- 2y + C, and with OT= 2 , [^^= I log (1 +a;3) + C . J 1+x^ 3 When ra = 4 , m may be , or 1 , or 2 , or 3 ; and the formula gives — with m = , /, '^^ =J_log*i±^^^ + J_tan-^^ + C: l+K* 4n/2 x^-xj2 + l 2^2 l-a;2 and with m = 1 , f^=C-Jtan-l; and with m = 2, f^=-Llog-^--f+l+-Ltan-'^4.C; yi+5c* 4^2 a;2 + a;^/2 + l 2 v/2 l-a;^ and with to = 3 , III. ALGEBRAIC FUNCTIONS — APPENDIX N. 181 j(l-ax)» 3a2 ^ ' f ^^ - C + A~^ i(a;H-l)^'52^1 V a'+l" (^—J^^-^-^ = c - /^ii i(a;-l)V52^ Va;-r {J'^jL^dx =C+ V(a: + «)(« + 6) + (a - &) log { \/a: + a + >/» + &}. ■^ <^)-/(P^>=«-;^''-(f)- B (13). Another form applicable whether a be + or - : ' with K.=x + 2a ( dx r dK )(ax^ + bx + cy J(aX^-|!+cy 182 TABLE OF INTEGRALS, IV. 1-9. IV. LOGARITHMS AND EXPONENTIALS. 1. je'dx = C + e'(e=base of Neperian Logarithms). 2. [a'"dx =0+ / a". J nlog,a Sp. Case : je'^dx = G+-e". 3. / log^dx =C + X logeS! - X . 4. jlog^dx =C + a;(logja!-log6e). Sp. Case: 6 = 10, ilog^gXdz = G + x{log^f^x- -4:34:29 ). 5. / {log^ydx = C + a:{(logea;)" - nQogfi;)"-' + n(n - l)(logea!)''~'' ±n\}. 6. jx'^e'dx = C + e'lx^ - mx'"-'^ + m{m - l)a;"'-''- ±m\). 7. id'^t'dx = is^'^h-'dx. (See III. A. 2.) For other formulse involving logarithms or exponentials, see VIII. 7-25 and IX. C. Definite Integrals loith Numerical (or Particular) Limits. 8. ("e-'^dx =1 /^ where a>0. Jo 2V a . Aog i Xdx = e-Vdx = T{n+ 1) . (See X. 1-6.) J a Jo 9 Appendix 0. IV. (7). From the equality of the logarithms of the two sides, a""'«6'' = »"'"%" and e"'°«?"=a!". Therefore fa" „nlog a+l n log^a + 1 and \e"^''^^dx =- ; n+\ Also I e-«a;''x + q). J p 2. / cosec^Qja; + q)dx = C - — cot (px + q) . J p q fe,m{px + q)dx ^ , 1 , , v J C0B^{px + q) p ^ ' , [cos (px + q)dx n 1 / , \ J 8ia'{px + q) p \-c- , 1/ Js. r = C + — loge tan (px + q) . sin (px + q) cos (px +'q) 6. [sm''xdx, Lo&^xdx, (-^, (-^, (tm''xdx, Icof'xdx. J J jsin"*' icosV J ' J For the first two integrals use II. F. or IX. B. 1, 2. „ „ second pair „ IX. B. 3, 4. „ „ last pair „ VI. 21, 22, See also VI. 19. 17. I sin (px + q) cos"(j3a; + q)dx = G- -. — - cos"+i( pa; + q) . J (n + V)p 18. I cos (px + q) sin''(pa; + q)dx = G + ■-. ^y- saiP''^\px + q) . 19. / sin"a; cos^ar^ia; . Four methods. Method I. (1) if m is an odd positive integer, use X=cosa! (2) „ n „ „ „ „ X=sina; (3) „ »H- w „ even negative „ „ X^tana; and the integrals become rational. Method II. If m and n are positive integers, use II. F. Method III. Use IX. B. 5, 6, 7 or 8. Method IV. If in or n or both are fractional, use Xssina; or=cosa;, and expand the binomial factor which results. 186 VI. TRIGONOMETRICAL FORMS, 20-25. 20. I sin™!!! cos"a; smqxcosrx dx. Where m, n, q, r are positive integers, use II. F. 21. jtaD.''xdx. If n IS even, = C+ — H + tana; + a;. M- 1 ra- 3 „ „odd =0+ =-- ^+----±-s— ± logecosa;. n- I n-3 2 22. (coVxdx = C ^ cof-ia; + -i- cof-^a! ± cot a; + », J n-1 n—3 -J. 11 n even. = C - — !^ cof-iw + -1- cot"-»a! - ± i cot^a; 71-1 n-3 +iog^sina;, ifModd. 23. f f =C+ J ,,cosh-i^^gg^±J if6^>a^ 7 a + CDS a; ^(o^ - a^) a + o cos a; CI _i a cos iE + & -f J.O ^ _o + —77-5 — jtrCos 1 J if 626, m+l/^^^' '''''' a cos x + b dx a + b cos X Definite Integrals with Particular \or NumericdC\ Limits. 26. / &m:'xdx=\ eoa"xdx=- — -. ^- (See X. 1-6.) i. ^0 2r(| + i) p /m + 1 \ p/ w+l \ „ /■'^'2 V 2 / \ 2 / 27. I sin"a;cos"a;(ia; = /^ . ^ , o\ — ' " " 188 VL TRIGONOMETRICAL FORMS — APPENDIX Q. Appendix Q. — Some other Special and some more General Cabbb. VI. (7) and (8). /x 1 sin.2 {px + q)dx = C + ^- - -^sin 2(px + q) , 2 4p /x 1 cos' {pz + q)dx = C + -^ + -r-sin ^{px + q) . 2 4p VI. (17) and (18). I sin (pa + q) cos {px + q)dx = C + — sin' (px + q) = C - ;j- cos 2{px + q) . ip /" • / . \ / .7\j r^ COS i(p + r)x + q + k} j sm {px + q) cos {rx + k)dx'= C ^''\^. ' — ^ '- cos {(p-r)x + q-k} f sin (px + q) sin (rx + k)dx = C - ^^ {(P + r)x + q + k} J 2{p + r) sin {{p-r)x + q-k] 2{p-r) f cos (px + q) cos (rx + k)dx = C + ^^^ {(P + r)x + g + k} J 2(p + r) sin {(p-r)x + q-k} 2(p-r) j sinpx sin (px + q)dx = C + -^ cos 3 - i sin (2pa) + g) . VI. (26) and (27). /•W2 . r,r/2 /-^ I sinaw[a;=l=l cosa;aic. / sva.xdx = 2. Jo Jo Jo j oosxdx = 0= j Gosxdx= I sinxdx. I siii^xdx = ~= r cos^xdx. Jo 2 J /^IP . „ , . , _ Mp g sin''(px + q)dx= — =j^ cos'' (px + q)dx . VI. TRIGONOMETRICAL FORMS — APPENDIX Q. 189 It p = ld and r — md, I and m being integers : that is, if d be the greatest common factor or divisor of p and r, I and m being calculable by — = i- : then p . — = l.2v and r . —^ = m . Ti-ir. m r d d Therefore, since iir = 360°, the addition of — - to a; in any com- d posite trigonometrical function of hofh (px + q) and (rx + k) brings the function recurrently back to the same value. It also does the same to any similar function of {(p + »')a; + constant} or of {(p -»•)» + constant}. Therefore the definite integration be- tween any value Kj and aij + — - of each of the three functions given above, VI. (17) and (18), namely of sin( ) cos( ), sin ( ) sin ( ), and cos ( ) cos ( ), gives zero integral. Also I sin (px + q) cos (px + q)dx = 0. If X denote flux of time, then -t=- is the lapse of time, or d "period," between successive recurrences of identical values of any composite trigonometrical function of (px + q) and (rx + k), corresponding to the " beats " of the composite harmonic function. The two harmonic functions have the different periods — and p 1ir 190 TABLE OF INTEGRALS, Vn. 1-5. VII.— INVEESB FUNCTIONS. [Note. — These can be transformed into mtegra,ls involving the corresponding direct functions by substitutions like X=sin~'a; .'.a;=sinX .\dx=cos'KdX, etc.] N.B.—smh-^x = log,{a: + ^(1 +x^)}. cosh-'a: = loge{a;± J{x^-l)}; x>l. tanh"'a; = -;7-loge= — - ; x\. Jt X — I secb-'a; = cosh-'— = log,-^- V(^~"'^); x0 X X , 1- J(\+x^) ., „ = log, ^^ ^, if «< . X gA~^z — sech~'cos x = tanh~'sin x = sinh~Han x = 2 tanh-Han |- = log.tan(^^ "'' y) = "2 ^"^'T^ 1 , 1 + sin a; sina' 1 . / sin-'a; and q are positive integeis. By II. F. and II. A. reduce to IX. C, 1 and 2. 14. l'E(x)f{siax,coax)dx, [¥() and/ ( )=rational integral •' functions]. By II. F. and U. A. reduce to IX. C, 1 and 2. r all 15. I log,(sin mx)dx = C — logj2 - k" (sin 2?wa; +52™ *»»« + -^sm6mx + -r^Bin87rKC + ) if oPdx = T{n+\). (See X.) 21. JVaog«rc?« = (-ir^|^,. (SeeX.) I 24. r 2??^cZa;=oo . J. "" 25. I log,(sin a!)rfa; = - -2-log,2 . Note. — Bernoulli's Numbers. Bi Ba Bj B4 B5 B(j B7 Bj B9 Bio ^to., etc. 1 1 1 1 _B_ B»l 7 8817 4 8867 12 2 277 T ^IF TJ Tff 15T ^TFff t SIO ■"?»! "4810 • 23. ?E^da;=^ if^>o a: .^ TABLE OF mTEGEALS, IX. A. 1. 195 IX.— rORMULiE or REDUCTION. Note. — Formulas of Reduction may be obtained by combinations of 1. 4, I. 7, and H. A, B, C. fiB^X'd IX. A. Algebraical. rdx where Xsacc" + h m, n, r being + or - , whole or fractional indices. By the use of one of the subjoined formulas, the integral of any one of the following 9 functions may be reduced to that of any of the other 8 : ^w+n^r+l ^mXr+1 ^_„Xr+l ^.m+nX' aTX' a-m-^X' gm+n-^r-1 a,mX'-l ^m-nX'-l . (i) is useful when m and n are of opposite sign ; (iii), (iv), (v) are useful when r is - ; (iii) and (vi) used together give the reduction from a;'"X'' to x'^'K^~^ (iv) „ (vi) „ „ „ „ „ „ „ „ a;™-»X'-^ (iv) „ (viii) „ „ „ „ „ „ „ „ x^-'^-^'-K N.B. — When r is a + integer, this integral can be dealt with by binomial expansion of X''. In other particular cases the sub- stitutions of III. A. 15 and III. A. 20 may be used. (i) (aj'-X'-^a; = ,— , ^ i a!™+iX'-+' -{m+\+nr + n)aj x'^-^^X^'dx | (ii) = , =^ ^ i a;™+i-"X'-+i -(m+l-n)b (x^^-'X^dx \ {m+l+nr)a\ ^ 'J I (iii) = / — ^^Tw; -f - aj^+^X'+i + (m+l +nr + n) (x'^X'+^dx \ ' («r + l)o( ^ '3 J ^'^^ = (m + l)&2 + ^r+l)62 { ^'""''■'"X'+^ _ ( w+l+ror + w)(m + 1 + nr + 2w) L™+„xr+,^ 1 OT+1 ] ] (v) = -. , \. I a!'»+i-"X''+' - (?» + 1 - w) f a;™-"X''+'*B I ^ ' «(r+l)a t ^ 'y j (vi) = \ I K^+'X*" + «r6 lx"'X'-''dx \ 196 IX. FOEMUL^ OF REDUCTION, A. 2-4. (vii) jafX'-dx = — i^ | ai^+'X' - nra j ar+"X'-^dx \ (-rai) = .— -^p- ,/ , - r / (w + 1 - M)a;'^iX'- (wH- 1 +TOr)(m+ 1 +Mr - w) ( ^ •' + ^af«-"X'-+'- {m+l-n)nr—lx'^-"X'-'dx \ 2. X=(aa;'» + &i;» + c) (i) [x^X'-dx = ^^ { w-^+iX*- + wrc iaTX'-^dx J m+l+nr I y -nraix'^+'^X^-'^dx \ - (»w + 1 + mr - m)6 \al^''X''dx | - (»» + 1 - 2«)c (x'^-^^X'^dx J 3. X=aa;2 + fee + c j-x'^X-idx = ^ I a!"-'X* - (ot - 1)6 jaf-^X-idx -{m-iyJar-'X-idx i If wi be a + integer, this reduces to III. B. 14 and 13. 4. X=ax^ + bx + c /^"^^ = (r-l)(4'ac-6^) |(^^^ + ^)^"'-" + 2(2r-3)a[x-'-+i£Za!l If r be a + integer, this reduces to III. A. 16, 17. „ „ (+ integer +^) „ „ III. B. 13. IX. FORMULA OB BBDUCTION, B. 1-6. 197 IX. B. Trigonometrical. Note. — The following formulae remain true when px + q is suhstituted for x and pdx for dx. (See II. G.) Sp. Case : If ^ = - 1, and q—^ radians, in this substitution, we deduce a new formula in which each trigonometrical ratio is replaced by its complementary ratio. N.B. — The following formulae, when n and m are integers, reduce bo the formulae referred to in the right-hand column. 'xdx 1. Isin''xdx= sin"-'a;cosa! + ^ /sin" ': J n n J 2. I cos'^xdx = — cos""^*; sin x + / (Ms''~''xdx J n n J o f dx - cos X 71-2 [ dx JsiD"x ~{n-l) sin"-'a! n-lj sin"-** . f dx _ sin a; a. ** " ^ f _ jcos"a! ~ (n-1) cos""^a; w-ljc 5. ^,n=js n-2 f dx sin™a;cos"aj(^a!. -"-771, n 1 sin^+'a; cos""'a; + - m + n sin*"-!* cos"+'a; + ^,„ m+n m+n n-\ m + n m-\ -*^^ii, n—'i -^^-2, re 6. X, J COI sm™a; ' cos"a! 1 sin^+'a) w - m - 2- w-1 '^x TO - 1 ■^"•■" w-1 cos»-'a! m-n cos""'a! m-ri Xm-2,n ) VI. 1, or j III. A. 1. 1 VI. 2, or / III. A. 1. Ivi. 6,orl2. I VI. 5, or 11. VI. 1, 2, or - 17, 18, or III. A. 1. VI. 1, 3, 5, or 16, or, III. A. 1. 198 IX. FORMULA OF KKDUCTION, B. 7-9. 7. X, dx Y _ 1 cos"+'!i; m - M - 2 Y Sin" 'a; m - 1 1 cos''"'a; w - 1 . ciTi'*~"ll« n Jsi n — m sm"~'a! w — m v sin"^!!; cos"a; VI. 1, 2, 4, or 16, or III. A. 1. Y _ 1 1 _ w + to-2y 1 m - 1 sin*""'!*; cos""'* »» — 1 .n-l-^'*' m + n 2, .VI. 5, 6, 15. 9. /tan"a;da; = *^°-" ''^ - Aan"-^i»(fo; . la;"! / IX. tORMUL.^ OF REDUCTION, C. 1-11. IX. C. Mixed Functions. of sin mxdx = - — cos mx + m x^ cos mxdx =^ —sin mx m m sin mx.dx mx IL(x''~h mj (x^-H sin mx m f cos mxi {n,-l)x''-^'^n~-l} x"-^ cos mxdx sin mxdx mxdx •n-l sin mxdx 199 TI. 1, or 2, when TO is a positive in- teger. Other wise, use II. D. ' and II. A. If » = a posi- tive integer J- VIII. lor 2. Otherwise, use II. D. HL A. 2 if f COS mx _ COS mx m f sin n r ^m+l « /■ 1 IIL A. 2 if 5. \(XogxYx'^dx = ^^aQ0xY ^\(\osxf-^^dx U=apo8i- 6. fx^a'"dx =4^ ^{yf'-'ard: J n log^a w logefly 7 fi!^ -g' 1 fe'dx ia;™ ~(»w-l)a!*-'"''m-lja;"-i o fax nj 6*^008" Ww sin a; -fa cos a;) 8. / ^"cos'^xdx = ^ = '- J n' + a' n(n-l) f ' n^ + a' j' + - e^'cos^-^xdx . \ IV. 2 if m= } a positive •* integer. ) VIII. i if m f is a whole ' number. IV. 2if»= even positive integer. VIII. 12if»i = odd positive integer. Q /^ ai ■ n J _ e'"sin"~^:g(a sing - wcos x) ] IV. 2 if M is an even y.je smxax n^ + a^ I positive integer. «(«^ r f VIII. 11 if ^^odd rt' + a^J ■ J positive integer. 10 ( ^ d ^{aoosx -{n- 2)sina;} a^ + ( n- 2f f e'$x ■ ] cos"* (re - 1)(« - 2) cos"-ia; (w - 1)(« - 2) j oos^V n r ^ A _ e°'{asing -t- (?» - 2)cosig} a^ -f (w - 2)^ /" e^tgg jsiH^ ~ (m - l)(m - 2)sin'-'«! ^ {n- l)(w- 2)/sin"-V 200 X. GAiSMA. CONC*IOSS, 1-6. X.— GAMMA FUNCTIONS. Properties of thb Gamua Function. 1. Definition: r(n)=|^ e-'x''-^dx= j ^ (log,^)"'^'^ where n>o. 2. r(M+l) = TOr(»t). 3. If re is a positive integer r(n) = (w - 1) ! and r(l) = 1. 4. r(re)r(l-«)=^ smWTT it n>o and 1 and<2, use the table. If n>2, by using (2) make the value depend on one in which n hes between 1 and 2. Thus r(3-52) = 2-52 x 1-52 r(l-52). If ml : thus T(n) = ^(n+l) ^ , r<-,„, r(1^63) — Example : r(-63) = .gg ' • XI. DUTERENTIAL EQUATIONS, A. 1-7. 201 XI. DIFFERENTIAL EQUATIONS. XI. A. First Order, First Degree. 1. X' + toX = 0, X = Ce-'», or a;= - — logS-. m ° G 2. X' +f{x) = , X = C - lf{x)dx . 3. X' + X/(a;) = , X = C eay { - h(x)dx) . Sp. Case : f{x) = mx^ ; log ^ = ^Ij^"*' . 4. X' + X/(a;) = <^(a;) X = exp{ — \f{x)dx}\G + j(x)exp{ jf{x)dx}dx] . Sp. Case : f{x) = Jc, .:X' + KK = ^{x); X = e-'"{G+ U{x)^dx}, 5. X' = 'l>{x)f{X) , j-^ = j{x)dx . Sp.Ca^e: 4>{x) = l, .•.X'=/(X), «=-j^y 6. /(»,X)X' + <^(a;,X) = . If the condition M, = ^ is fulfilled, then j,t>{x,X)dx + j[f{x,X) - ^^^^dx'^X = C or jf{x,X)dX + [[^(aX) - J^^M)axJrfx = C the integrations being partial. Note. — If the equation as given does not fulfil the above con- dition, it may do so after being multiplied throughout by a function of a: or X or both, called the Integrating Factor. (See Boole, chapters IV., V.) E.g. {x^X+X + l) + X'{x + x^). Integrating Factor, 1/(1 + x^). Solution : xX + tan"^a! = C . 7. Xy(a;,X) + <^(a:,X) = where /(a;,X) + <^(a;,X) is homogeneous in x,X. Substitute X^a;^ and reduce to XI. A. 5. 202 XI. DIFFERENTIAL EQUATIONS, A. 8. (ax + bX + c)X' + {/x + gX + h) = 0. Assume ax + bX + e=z ; fx + gX + A=Z , hence - Z'z +/z - jfZ = . (See XI. A. 7 or 3.) Mee. when a:b =f:g , put S=ox + bX , hence (J + <=){S -a) + gM + bh = 0. (See XI. A. 6 Sp. Case.) 9. X' + X/(x) = X''.^(a;). Substitute 4^=Xi-" . Then S + (1 - n)Mf{x) = (1 - n)4,ix) . (See XI. A. 4.) 10. If a;X' = (AX + B)(aX + 6); then Ca;" = ^= — =- , where a = Ab~ aB. aX + 6 If xX' = aX^ + bX + c; then ^'^'' = 2aX + b + 1 ' '^^^^^ "■= JW^^Iae, 11. If a;X' = (AX + B)(aa) + 6); thenAX + B = a;*».e*"*'+'". Xr. DIFFERENTIAL EQUATIONS, B. 203 IX. B. First Order, Second or Higher Degree. 1. /(a;,X,X') = 0. If possible, solve for X'. Each solution X' = <^(a;,X) , solved by XI. A. if possible, gives part of the general solution. 2. /(X,X') = . Use XI. B. 1 if possible : otherwise solve for X if possible. Each solution X = '(S.')dX' + C . Then eliminate X' between the last two equations. 4. X = a;X'+/(X'). (Clairault's Equation) X=:ex+f{c). See §202, p. 123. 204 XI. DIFFERENTIAL EQUATIONS, C. XI. C. Second Order. 1. X" + ??i2X = 0, X = A cos ??ia; + B sin TTia: » = cos (mx + K). 2. X"-m2X = 0, X = Ae"" + Be-"«. 3. X" + a X' + 6X = 0. Two forms :— X = e— /^ I Aea;p(|- J(a^ - 46)) + Beay ( - 1. ^(a^ - 46)) i when a2>46 -- { Acos(|vIF3H-^).Bsin(|v463^.) } ) ^^ = Ce-«/^cos||-7(46-a2) + Ki j <**• 4. X" =/(a;), X = j j f{x)dxdx + Aa; + B . 6. X" + aX' + 6X =/(«). By XI. C. 3, find M by solving X + a3^' + b^ = 0. „ XL A. 4, „ S „ „ 3^S' + {2j' + aJ}B=/(a!), Then X = 4!f fHt^ic . See § 2 1 6, page 132. 7. X" + X'/(a;) + XF(a;) = .^(a;). Find M if possible from J" + M'f{x) + JF(a;) = 0. „ B by XI. A. from n'M + 'B{2j + Mf{x)} = {x). Then X = ^j'adx. See § 2 1 4, page 1 30. /72Y /-/SY 8. — 5 = c^ — s . ■where X is a function of x and y : General integral solution : — where the forms of the functions /( ) and <^( ) are deter- mined by limiting conditions. XI. DIFFERENTIAL EQUATIONS, D. 205 XI. D. Order higher than Second. 1. f(x, X'"-", XW) = 0. Put J = X'"-", and equation becomes /{x, ^, J') = 0. (See XI. A, or B.) 2. f(x, Xf-^ X"-^', X<"1,) = 0. Put J = X"-^' and equation becomes f{x, J, J', J") = 0. (See XI. C.) 3. XI") =/(»). X = / f{x)da!" + C^x"-^ + Cja;"-^ + - - - - + C„ . 4. X"*' + aiX<"-« + agX'"-") + - - - + a„. ^X' + a„X = where wij, m^, m„ are the roots of the auxiliary equation OT" + ai?w"-i + a„??i"-2 + + a„_jm + a„ = 0. Note 1. — If wij =j? + q J -I and wij =j? -q ^ -I are a pair of imaginary roots, the terms Gj^^" + C^e^^ are equivalent to the real form ^' (A cos ja; + B sin g's;). Note 2. — If there be r equal roots m^, m2 m„ each=/i, the corresponding terms in the value of X are (Cj + G.^ + G^a? H + G'iif~^)ei^. And if there are r pairs of imaginary roots each —P ± ?>/ ~ 1) *li6 terms are e*""! AjCOS qx + BjSin ya; + ^(Ajcos qx + Bjsin qx) H + a!''~^(A^cos qx + B,sin ga;)} 5. X"" + aiXi-" + a2X'"-=" + — - + a„X = 6„ + Mh- 622;^+ — - + h^. Differentiate both sides r + 1 times and solve the resulting equation by XI. D. 4. This solution is too general, having n + r+\ arbitrary constants : but by substituting in the equation and using I. 11, we get r + 1 relations between the constants. Otherwise : see XI. D. 6. 206 XI. DIFFERENTIAL EQUATIONS, D. 6. X"" + aiX'"-ii + a^Xi"-^' + - - - + a„X =f{x.) Let X = F(a;) be tte solution on the supposition that /(a;) = 0. (SeeXI. D. 4.) Then X = F(a;) + 2 A^p{m^) j exp{ - m^f{x)}dx, where Ai Aj are such as to make the equation Ai A2 A„ _ 1 ■m-7?ii ' TO-9W.2 ' ' »i-wi„ m" + fflim""' + asm""'' H i-a„ identically true. (See II. E.) Another method : By variation of Parameters j see Forsyth, §75. XI. DIFFERENTIAL EQUATIONS, E. 207 XI. E. Partial Differential Equations. 1. ! = «'§. y = Gexp{a.x + o?aH) where C and a are arbitrary constants, or 2/ = { A cos aa; + B sin ax}exp{ - a?aH), A and B being arbitrary constants. Oenerdl Solution : y = sum of any number of solutions like the above. 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" No more enduring Memorial of Professor Rankine conH be devised than the pubUc. don of these papers in an accessible form. . . . The OjUection is most valuable o._^ ac^unt ol the nature of his discoveries, and the beauty and completeness of his analysis, — Architect. lONDOM: CHAHLES GRIFflN & CO., LIMITED. EXETER STREET, STRAND.- 36 CHARLES ORIFFIN ds GO.'S PUBLICATIONS. Third Edition, Thoroughly Revised and Enlarged. With 60 Plates and Numerous Illustrations. Handsome Cloth. 34s. HYDRAULIC POWER AND HYDRAULIC MACHINERY. BY HENRY ROBINSON, M. Inst. C.E., F.G.S., FELLOW OF king's COLLEGE, LONDON ; PROF. 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With Practical Illustrations of Enoines and Pumps applied to Hinino, Town Wateb Supply, Dkainage of Lands, &c., also Economy and Efficiency Trials of Pumping Machinery. Bt heney davey, Member of the InBUtntion of GiTil Engineers, Member of the Institution of Mechanical Engineers, F.&.S., &o. OoNTENTS — Early History of Pumping En^es — Steam Pumpmg Engines— Pumps and Pump Valves — General Principles of N'on-Botative Pumping Engines — The Cornish Engine, Simple and Compound — Types of Mining Engines— Pit Work — Shaft Sinking — Hydraulic Transmission of Power in Mines — Electric Transmission of Power — Valve Gears of Pumping Engines — Water Pressure Pumping Engines — Water Works En^es — Pumping Engine Economy and Trials of Pumping Machinery — CentnfUgal and other Low-Lift Pumps — Hydraulic Rams, Pumping Mains, &c. — Index. ''By the *one English Engineer who probably knows more abont Pumping Machinery than ANY OTHBB.' ... A VOLUUE BBOOBDING THB BESULTS OF LONG BXPBBIBNCE AND STUDY." — The Engineer. "Undoubtedly THE BEST and most pbactigai. treatise on Pumping Machinery that has TET BEEN PUBLISHED." — JUiniTig Journal. UNDON: CHARLES GRIFFIN & CO.. LIMITED, EXETER STREET. STRAND NAVAL ARGHirEOTURE. 37 At Press. In Large 8vo. Handsome Cloth. Profusely Illustrated. In Two Volumes, Each Complete in itself, and Sold Sepakately. THE DESIGN AND CONSTRUCTION OF SHIPS. By JOHN HARVARD BILES, M.Inst.N.A., Professor of Naval Architecture in Glasgow UniTersity. Contents of Volume I. — Part I. : General Considerations. — ^Methods of Determin- ation of the Volume and Centre of Gravity of a known Solid. — Graphic Rules for Integration.— Volumes and Centre of Gravity of Volumes.— Delineation and Descriptive Geometry of a Ship's Form.— Description and Instances of Ship's Forms.— Description of Types of Ships. Part II. : Calculation of Displacement, Centre of Buoyancy and Areas.- Metacentres.— Trim.— CoefBcients and Standardising.— Kesults of Ship Calcula- tipns. — Instruments Used to Determine Areas, Moments, and Moments of Inertia of Plane Curves.— Cargo Capacities,— Effects on Draught, Trim, and Initial Stability due to Flooding Compartments. — Tonnage.— Freeboard. — Launching.— Application of the Integraph to Ship Calculations.— Straining due to Unequal Longitudinal Distribution of Weight and Buoyancy.— Consideration of Stresses in a Girder.— Application of Stress Formula to the Section of a Ship.— Shearing Forces and Bending Moments on a Ship amongst Waves.— Stresses on the Structure when Inclined to the Upright or to the Line of Advance of the Waves. — Distribution of Pressure on the Keel Blocks of a Vessel in Dry Dock.— Consideration of Compression in Ship Structure. BY PROFESSOR BILES. LECTURES ON THE MARINE STEAM TURBINE. With 131 Illustrations. Price 6s. net. See page 28. Royal Suo, Handsome Cloth. With numerous Illustrations and Tables. 25s. THE STABILITY OP SHIPS. BY SIR EDWARD J. REED, K.C.B., F.R.S., M.P., CHIGBT 0» THK IMPBRIAL ORDERS OF ST. STANILAUS Of RUSSIA.; FRANCIS JOSKPB OF AUSTRIA ; MBDJIDIE OF TURKEY ; AMD RISING SUN OF JAPAN ; VICF- PRESIDKNT OF THE INSTITUTION OF NAVAL ARCHITECTS. " Sir Edward Reed's ■ Stability of Ships ' is invaluable. The Naval Architect will find brought together and ready to his hand, a mass of information wbich he would othtt- •wise have to Mek in an almost endless variety of publications, and some of which he wmild possibly not be able to obtain at all elsewhere."— iS<«i»«fey. ILOMDON : CHARLES GRIFFIN & CO.. LIMITED, EXETER STREET. STRAND, 38 yJHARLES aRIFFIN Ji OO.'S PUBLICATIOSS. WORKS BY THOMAS WALTOTf, NATAL ARCHITECT. Third Edition. Illustrated with Plates, Numerous Diagrams, and Figures in the Text. i8s. net. STE EL SHI PS; THEIR OOlSrSTRUCTION AND MAIITTENANCB. 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Altogether, the work is excellent, and will prove of great value to those for whom It is intended.*'— 7%e Engineer. In Large 8vo. Handsome Cloth. With 2.S6 Pages, 40 Folding and 9 other Plates, and very numerous Illustrations in the Text. 7s. 6d. net. PRESENT-DAY SHIPBUILDING. For Shipyard Students, Ships' Officers, and Engineers. Bt THOS. WALTON. Gbneral Contents. — Classification. — Materials used in Shipbuilding. — Alternative Modes of Construction. — Details of Construction. — Framing, Plating, Rivetting, Stem Frames, Twin-Screw Arrangements, Water Ballast Arrangements, Loading and Discharging Gear, &c. — Types of Vessels, including Atlantic Liners, Cargo Steamers, Oil carrying Steamers, Turret and other .Self Trimming Steamers, &;c. — Index. NunE Edition, Illustrated. Handsome Cloth, CrcyumSvo. 7a.6d. The Chapters on Tonnage and Freeboard have been brought thoroughly up to date, and embody the latest (1906) Board of Trade Regulations on these subjects. KNOW YOUR OWN SHIP. By THOMAS WALTON, Naval Architect. Specially arranged to suit the requirements of Ships' Officers, Shipowners Superintendents, Draughtsmen, Engineers, and Others, CONTEHTS. — Displacement and Deadweight. — Moments. — Buoyancy. — Strain. — Structure. — Stability. — Rolling. — Ballasting. — Loading. — Shifting Cargoes. — Effect of Admission of Water into Ship. — ^Trim Tonnage.^fVeeboard (Load-line),— ValculationB, — Set of Calculations from Actual Drawings.— Ibhex. " The work iB of the highest value, and all who go down to the sea In ships should make them- selves acquainted with it.'— Shipping World (on the new edition), lONDOM: CHARLES GRIFFIN & CO., LIMITED, EXETER STREET. STRAND. NAUTICAL WORKS. 39 GRIFFIN'S NAUTICAL SERIES. Edited by EDW. BLACKMORE, Master Mariner, First Class Irinity House Certificate, Assoc. Inst. N.A. ; Ahd Weitten, mainly, by Sailom for Sailoks. "This admieablu series. "—J'airpZay. "A tert dsbful series."— jro(ttr«. 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LONDON: CHARLES GRIFFIN & CO., LIMITED, EXETER STREET, STRAND. 40 GBARLES GRIFFIN . '* This admibable uahual, by Oapt. Wilson-Babbxb of the ^ Worcester,' seems to uft PBBFBCTLT DESIGNED, and holds its place excellently in ' Gbiffib's Nautioal Sbbibs. ' . Although intended for those who are to become OflBcers of the Merchant Navy, It will be round useful by all TACHTSUBN.".~^fAma9ufn. **'" For complete List of (^eivfot's Nautical Series, see p. 39. LONDON : CHARLES CRIFFIN & CO., LIMITED. EXETER STREET, STRAND. NAUTICAL WORKS. 41 GRIFFIN'S NAUTICAL SERIES. Second Edition, Bevised cmd Illiistrated, Price Ss. 6d. By DAVID WILSON-BARKER, R.N.R., F.R.S.E., &o., &o., AND WILLIAM ALLINGHAM, FIBST-OLASS HOHOUBS, NATiaATION, SOIENOS AND ABT DEPAKTMENT. Wlitb filumerous illustrations and Bsamination (Questions. . General Contents. — Defimtiona — Latitude and Longitude — Instrumeuta of Navigation — Correction of Courses — Plane Sailing — Traverse Sailing— Day's Work — Parallel Sailing — Middle Latitude SaUing — Mercator's Chart— Mercator Sailing — Current Sailing — Position by Bearings — Great Circle Sailing —The Tides — Questions — Appendiz : Compass Error — Numerous Useful Hints &;c. — Index. '* Fbbgiselt the kind of work required for the New Certificates of competency in grades Irom Second Uate to extra Master. . . , Candidates will find it iitvaltjable. "—i)unde/ Adeertiter. " A OAFTFAL XJTTLB BOOK . . . Specially adapted to the New ExaminatioiiB. The Authors are Oapt. Wn.aoiT-BAKKBB (Captaia-SuperintendeHt of the Nautical College. H.M.S. ' Worcester,' who has had great experience in the highest problems of Navigatioa), and Ub. Aujnghau, a well-known writer on the Science of Navigation and Nautical Astronomy. " —Shipping World, Handsome Cloth. Fully Illustrated. Price 7s. 6d. MARINE METEOROLOGY, FOR OFFICERS OF THE MERCHANT NAVY. By WILLIAM ALLINGHAM, Joint Author of "Navigation, Theoretical and Practical." With numerous Plates, Maps, Diagrams, and Illustrations, and a facsimile Reproduction of a Page from an actual Meteorological Log-Book. STJMMAEY OP CONTENTS. Introductory. — Instruments Used at Sea for Meteorological Purposes. — Meteoro- logical Log-Books.— Atmospheric Pressure.— Air Temperatures.— Sea Temperatures.— Winds. — Wind force Scales. — History of the Law of Storms. — Hurricanes, Seasons, and Storm Tracks.- Solution of the Cyclone Problem.— Ocean Currenta.-Iceberga.— Syn- chronous Charts.- Dew, Mists, Togs, and Haze.— Clouds.— Kain, Snow, and Hail.— Mirage, Kainbowa, Coronas, Halos, and Meteors.— Lightning, Corposants, and Auroras.— QDESTIONS.—APPBKDIX.— Index. " Quite the best publication, and. certainly the most interestihg, on this subject ever presented to Nautical men."Shipping Gazette. *,* For Complete List of Gkifhn's Natttioal Series, see p. 39, lONOON: CHARLES GRIFFIN & CO.. LIMIFED. EXETER STREET, STRAND- 4* OBAKLBH GRIKFIN A OO.'t PUBLICATIOm. GRIFFIN'S XAUTICAL SERIES. Sboono Edition, Revised. With Numerous Illustrations. Price 3s. 6d. Practical Mechanics: Applied to the Requirements of the Sailor. By THOS. MACKENZIE, Master Mariner, F.R.A.S, Gemubai, Contbnts. — Resolution and Composition of Forces — Work done by Machines and Living Agents — The Mechanical Powers: The Lever; Derricks as Bent Levers— The Wheel and Axle : Windlass ; Ship's Capstan ; Crab Winch— Tackles : the "Old Man"— The Inclined Plane; the Screw— The Centre of Gravity of a Ship and Cargo — Relative ■Strength of Hope : Steel Wire, Manilla, Hemp, Coir — Derricks and Shears— Calculation of the CioBB-breaking Strain of Fir Spar— Centre of EfEort of Sails — Hydrostatics: the Diving-bell ; Stability of Floating Bodies ; the Ship's Pump, &c. " This excellent book . . . contains a large amount of information. " — Nature. " Well worth the money . . . will be found bxobedinoli helpitol."— Shhiping World. "No Ships' Oitioees' bookcase will henceforth be complete without Caitain Mackenzie's ' Peaotical Mbchanios.' Notwithstanding my many years' experience at sea, it has told me Jwvt much more there is to acquire." — (Letter to the Publishers from a Master Mariner). " I must express my thanks to you for the labour and care you have taken in 'Pbaotioal Mechanics.' . . . It la a life's experience. . . What an amount we frecjuently see wasted by rigging purchases without reason and accidents to spars, &c., &o. ! 'Practical Mbchanios' wonLD save all this." — (Letter to the Author from another Master Mariner). WORKS BY RICHARD C. BUCK, of the Thames Nautical Training OoUege, H.M.S, ' WorceBter,' A Manual of Trig^onometry : Vl/ith Diagrams, Examples, and Exercises. Price 3s. 6cl. Third Edition, Revised and Corrected. *,* Mr. Buck's Text-Book has been specially pebpaked with a view to the New Examinations of the Board of Trade, in which Trigonometry is an obligatory subject. "This BuiNBHTLY PBAOTicAL and RELIABLE VOLUME." — Schoolmaster. A Manual of Algebra. 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